On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry...

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On the Sobolev quotient in CR geometry Joint work with J.H.Cheng and P.Yang Andrea Malchiodi (SNS) Banff, May 7th 2019 Andrea Malchiodi (SNS) Banff, May 7th 2019 1 / 16

Transcript of On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry...

Page 1: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

On the Sobolev quotient in CR geometryJoint work with J.H.Cheng and P.Yang

Andrea Malchiodi (SNS)

Banff, May 7th 2019

Andrea Malchiodi (SNS) Banff, May 7th 2019 1 / 16

Page 2: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

The Yamabe problem

The Yamabe problem consists in finding conformal metrics with constantscalar curvature on a compact manifold (M, g). If Rg is the backgroundone, setting g(x) = λ(x)g(x) = u(x)

4n−2 g(x), u(x) one has to solve

(Y ) −cn∆u+Rg = Run+2n−2 ; cn = 4

n− 1

n− 2, R ∈ R.

Considering R as a Lagrange multiplier, one can try to find solutions byminimizing the Sobolev-Yamabe quotient

QSY (u) =

∫M

(cn|∇u|2 +Rgu

2)dV(∫

M |u|2∗dV

) 22∗

; 2∗ =2n

n− 2.

The Sobolev-Yamabe constant is defined as

Y (M, [g]) = infu6≡0

QSY (u).

Andrea Malchiodi (SNS) Banff, May 7th 2019 2 / 16

Page 3: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

The Yamabe problem

The Yamabe problem consists in finding conformal metrics with constantscalar curvature on a compact manifold (M, g).

If Rg is the backgroundone, setting g(x) = λ(x)g(x) = u(x)

4n−2 g(x), u(x) one has to solve

(Y ) −cn∆u+Rg = Run+2n−2 ; cn = 4

n− 1

n− 2, R ∈ R.

Considering R as a Lagrange multiplier, one can try to find solutions byminimizing the Sobolev-Yamabe quotient

QSY (u) =

∫M

(cn|∇u|2 +Rgu

2)dV(∫

M |u|2∗dV

) 22∗

; 2∗ =2n

n− 2.

The Sobolev-Yamabe constant is defined as

Y (M, [g]) = infu6≡0

QSY (u).

Andrea Malchiodi (SNS) Banff, May 7th 2019 2 / 16

Page 4: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

The Yamabe problem

The Yamabe problem consists in finding conformal metrics with constantscalar curvature on a compact manifold (M, g). If Rg is the backgroundone, setting g(x) = λ(x)g(x) = u(x)

4n−2 g(x), u(x) one has to solve

(Y ) −cn∆u+Rg = Run+2n−2 ; cn = 4

n− 1

n− 2, R ∈ R.

Considering R as a Lagrange multiplier, one can try to find solutions byminimizing the Sobolev-Yamabe quotient

QSY (u) =

∫M

(cn|∇u|2 +Rgu

2)dV(∫

M |u|2∗dV

) 22∗

; 2∗ =2n

n− 2.

The Sobolev-Yamabe constant is defined as

Y (M, [g]) = infu6≡0

QSY (u).

Andrea Malchiodi (SNS) Banff, May 7th 2019 2 / 16

Page 5: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

The Yamabe problem

The Yamabe problem consists in finding conformal metrics with constantscalar curvature on a compact manifold (M, g). If Rg is the backgroundone, setting g(x) = λ(x)g(x) = u(x)

4n−2 g(x), u(x) one has to solve

(Y ) −cn∆u+Rg = Run+2n−2 ; cn = 4

n− 1

n− 2, R ∈ R.

Considering R as a Lagrange multiplier, one can try to find solutions byminimizing the Sobolev-Yamabe quotient

QSY (u) =

∫M

(cn|∇u|2 +Rgu

2)dV(∫

M |u|2∗dV

) 22∗

; 2∗ =2n

n− 2.

The Sobolev-Yamabe constant is defined as

Y (M, [g]) = infu6≡0

QSY (u).

Andrea Malchiodi (SNS) Banff, May 7th 2019 2 / 16

Page 6: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

The Yamabe problem

The Yamabe problem consists in finding conformal metrics with constantscalar curvature on a compact manifold (M, g). If Rg is the backgroundone, setting g(x) = λ(x)g(x) = u(x)

4n−2 g(x), u(x) one has to solve

(Y ) −cn∆u+Rg = Run+2n−2 ; cn = 4

n− 1

n− 2, R ∈ R.

Considering R as a Lagrange multiplier, one can try to find solutions byminimizing the Sobolev-Yamabe quotient

QSY (u) =

∫M

(cn|∇u|2 +Rgu

2)dV(∫

M |u|2∗dV

) 22∗

; 2∗ =2n

n− 2.

The Sobolev-Yamabe constant is defined as

Y (M, [g]) = infu6≡0

QSY (u).

Andrea Malchiodi (SNS) Banff, May 7th 2019 2 / 16

Page 7: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

The Sobolev quotient in Rn (n ≥ 3)

Recall the Sobolev-Gagliardo-Nirenberg inequality in Rn

‖u‖2L2∗ (Rn)

≤ Bn∫Rn

|∇u|2dx; u ∈ C∞c (Rn).

As for Y (M, [g]), define the Sobolev quotient Sn = infu

∫Rn cn|∇u|

2dx

‖u‖2L2∗ (Rn)

.

Completing C∞c (Rn), Sn is attained by ([Aubin, ’76], [Talenti, ’76])

Up,λ(x) :=λ

n−22

(1 + λ2|x− p|2)n−22

; p ∈ Rn, λ > 0.

• Since Sn is conformal to Rn, one has that Y (Sn, [gSn ]) = Sn.

Andrea Malchiodi (SNS) Banff, May 7th 2019 3 / 16

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The Sobolev quotient in Rn (n ≥ 3)

Recall the Sobolev-Gagliardo-Nirenberg inequality in Rn

‖u‖2L2∗ (Rn)

≤ Bn∫Rn

|∇u|2dx; u ∈ C∞c (Rn).

As for Y (M, [g]), define the Sobolev quotient Sn = infu

∫Rn cn|∇u|

2dx

‖u‖2L2∗ (Rn)

.

Completing C∞c (Rn), Sn is attained by ([Aubin, ’76], [Talenti, ’76])

Up,λ(x) :=λ

n−22

(1 + λ2|x− p|2)n−22

; p ∈ Rn, λ > 0.

• Since Sn is conformal to Rn, one has that Y (Sn, [gSn ]) = Sn.

Andrea Malchiodi (SNS) Banff, May 7th 2019 3 / 16

Page 9: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

The Sobolev quotient in Rn (n ≥ 3)

Recall the Sobolev-Gagliardo-Nirenberg inequality in Rn

‖u‖2L2∗ (Rn)

≤ Bn∫Rn

|∇u|2dx; u ∈ C∞c (Rn).

As for Y (M, [g]), define the Sobolev quotient Sn = infu

∫Rn cn|∇u|

2dx

‖u‖2L2∗ (Rn)

.

Completing C∞c (Rn), Sn is attained by ([Aubin, ’76], [Talenti, ’76])

Up,λ(x) :=λ

n−22

(1 + λ2|x− p|2)n−22

; p ∈ Rn, λ > 0.

• Since Sn is conformal to Rn, one has that Y (Sn, [gSn ]) = Sn.

Andrea Malchiodi (SNS) Banff, May 7th 2019 3 / 16

Page 10: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

The Sobolev quotient in Rn (n ≥ 3)

Recall the Sobolev-Gagliardo-Nirenberg inequality in Rn

‖u‖2L2∗ (Rn)

≤ Bn∫Rn

|∇u|2dx; u ∈ C∞c (Rn).

As for Y (M, [g]), define the Sobolev quotient Sn = infu

∫Rn cn|∇u|

2dx

‖u‖2L2∗ (Rn)

.

Completing C∞c (Rn), Sn is attained by ([Aubin, ’76], [Talenti, ’76])

Up,λ(x) :=λ

n−22

(1 + λ2|x− p|2)n−22

; p ∈ Rn, λ > 0.

• Since Sn is conformal to Rn, one has that Y (Sn, [gSn ]) = Sn.

Andrea Malchiodi (SNS) Banff, May 7th 2019 3 / 16

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The Sobolev quotient in Rn (n ≥ 3)

Recall the Sobolev-Gagliardo-Nirenberg inequality in Rn

‖u‖2L2∗ (Rn)

≤ Bn∫Rn

|∇u|2dx; u ∈ C∞c (Rn).

As for Y (M, [g]), define the Sobolev quotient Sn = infu

∫Rn cn|∇u|

2dx

‖u‖2L2∗ (Rn)

.

Completing C∞c (Rn), Sn is attained by ([Aubin, ’76], [Talenti, ’76])

Up,λ(x) :=λ

n−22

(1 + λ2|x− p|2)n−22

; p ∈ Rn, λ > 0.

• Since Sn is conformal to Rn, one has that Y (Sn, [gSn ]) = Sn.

Andrea Malchiodi (SNS) Banff, May 7th 2019 3 / 16

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Brief history on the Yamabe problem

- In 1960 Yamabe attempted to solve (Y ) by subcritical approximation.

- In 1968 Trudinger proved that (Y ) is solvable provided Y (M, [g]) ≤ εnfor some εn > 0. In particular for negative and zero Yamabe class.

- In 1976 Aubin proved that (Y ) is solvable provided Y (M, [g]) < Sn.He also verified this inequality when n ≥ 6 and (M, g) is not locallyconformally flat, unless (M, g) ' (Sn, gSn).

- In 1984 Schoen proved that Y (M, [g]) < Sn in all other cases, i.e. n ≤ 5or (M, g) locally conformally flat, unless (M, g) ' (Sn, gSn).

Andrea Malchiodi (SNS) Banff, May 7th 2019 4 / 16

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Brief history on the Yamabe problem

- In 1960 Yamabe attempted to solve (Y ) by subcritical approximation.

- In 1968 Trudinger proved that (Y ) is solvable provided Y (M, [g]) ≤ εnfor some εn > 0. In particular for negative and zero Yamabe class.

- In 1976 Aubin proved that (Y ) is solvable provided Y (M, [g]) < Sn.He also verified this inequality when n ≥ 6 and (M, g) is not locallyconformally flat, unless (M, g) ' (Sn, gSn).

- In 1984 Schoen proved that Y (M, [g]) < Sn in all other cases, i.e. n ≤ 5or (M, g) locally conformally flat, unless (M, g) ' (Sn, gSn).

Andrea Malchiodi (SNS) Banff, May 7th 2019 4 / 16

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Brief history on the Yamabe problem

- In 1960 Yamabe attempted to solve (Y ) by subcritical approximation.

- In 1968 Trudinger proved that (Y ) is solvable provided Y (M, [g]) ≤ εnfor some εn > 0.

In particular for negative and zero Yamabe class.

- In 1976 Aubin proved that (Y ) is solvable provided Y (M, [g]) < Sn.He also verified this inequality when n ≥ 6 and (M, g) is not locallyconformally flat, unless (M, g) ' (Sn, gSn).

- In 1984 Schoen proved that Y (M, [g]) < Sn in all other cases, i.e. n ≤ 5or (M, g) locally conformally flat, unless (M, g) ' (Sn, gSn).

Andrea Malchiodi (SNS) Banff, May 7th 2019 4 / 16

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Brief history on the Yamabe problem

- In 1960 Yamabe attempted to solve (Y ) by subcritical approximation.

- In 1968 Trudinger proved that (Y ) is solvable provided Y (M, [g]) ≤ εnfor some εn > 0. In particular for negative and zero Yamabe class.

- In 1976 Aubin proved that (Y ) is solvable provided Y (M, [g]) < Sn.He also verified this inequality when n ≥ 6 and (M, g) is not locallyconformally flat, unless (M, g) ' (Sn, gSn).

- In 1984 Schoen proved that Y (M, [g]) < Sn in all other cases, i.e. n ≤ 5or (M, g) locally conformally flat, unless (M, g) ' (Sn, gSn).

Andrea Malchiodi (SNS) Banff, May 7th 2019 4 / 16

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Brief history on the Yamabe problem

- In 1960 Yamabe attempted to solve (Y ) by subcritical approximation.

- In 1968 Trudinger proved that (Y ) is solvable provided Y (M, [g]) ≤ εnfor some εn > 0. In particular for negative and zero Yamabe class.

- In 1976 Aubin proved that (Y ) is solvable provided Y (M, [g]) < Sn.

He also verified this inequality when n ≥ 6 and (M, g) is not locallyconformally flat, unless (M, g) ' (Sn, gSn).

- In 1984 Schoen proved that Y (M, [g]) < Sn in all other cases, i.e. n ≤ 5or (M, g) locally conformally flat, unless (M, g) ' (Sn, gSn).

Andrea Malchiodi (SNS) Banff, May 7th 2019 4 / 16

Page 17: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

Brief history on the Yamabe problem

- In 1960 Yamabe attempted to solve (Y ) by subcritical approximation.

- In 1968 Trudinger proved that (Y ) is solvable provided Y (M, [g]) ≤ εnfor some εn > 0. In particular for negative and zero Yamabe class.

- In 1976 Aubin proved that (Y ) is solvable provided Y (M, [g]) < Sn.He also verified this inequality when n ≥ 6 and (M, g) is not locallyconformally flat, unless (M, g) ' (Sn, gSn).

- In 1984 Schoen proved that Y (M, [g]) < Sn in all other cases, i.e. n ≤ 5or (M, g) locally conformally flat, unless (M, g) ' (Sn, gSn).

Andrea Malchiodi (SNS) Banff, May 7th 2019 4 / 16

Page 18: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

Brief history on the Yamabe problem

- In 1960 Yamabe attempted to solve (Y ) by subcritical approximation.

- In 1968 Trudinger proved that (Y ) is solvable provided Y (M, [g]) ≤ εnfor some εn > 0. In particular for negative and zero Yamabe class.

- In 1976 Aubin proved that (Y ) is solvable provided Y (M, [g]) < Sn.He also verified this inequality when n ≥ 6 and (M, g) is not locallyconformally flat, unless (M, g) ' (Sn, gSn).

- In 1984 Schoen proved that Y (M, [g]) < Sn in all other cases, i.e. n ≤ 5or (M, g) locally conformally flat, unless (M, g) ' (Sn, gSn).

Andrea Malchiodi (SNS) Banff, May 7th 2019 4 / 16

Page 19: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

On the inequality Y (M, [g]) < Sn

The inequality is proved using Aubin-Talenti’s functions. Given p ∈M ,

consider a conformal metric g ' U4

n−2

p,λ g with λ large. Since locally(M, g) ' Rn and since Up,λ is highly concentrated, QSY (Up,λ) ' Sn,with small correction terms due to the geometry of M .

Since Up,λ decays like 1|x|n−2 at infinity, it is more localized in large dimen-

sion. Aubin proved that for n ≥ 6 the corrections are given by − |Wg |2(p)λ4

,a local quantity depending on the Weyl tensor.

For n ≤ 5 the correction is of global nature. Heuristics: if u ' Up,λ then

Lgu := −cn∆u+Rgu ' Un+2n−2

p,λ '1

λδp.

At large scales an approximate solution looks like the Green’s functionGp of the operator Lg. If Gp ' 1

|x|n−2 +A at p, the correction is −A/λn−2.

Andrea Malchiodi (SNS) Banff, May 7th 2019 5 / 16

Page 20: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

On the inequality Y (M, [g]) < Sn

The inequality is proved using Aubin-Talenti’s functions.

Given p ∈M ,

consider a conformal metric g ' U4

n−2

p,λ g with λ large. Since locally(M, g) ' Rn and since Up,λ is highly concentrated, QSY (Up,λ) ' Sn,with small correction terms due to the geometry of M .

Since Up,λ decays like 1|x|n−2 at infinity, it is more localized in large dimen-

sion. Aubin proved that for n ≥ 6 the corrections are given by − |Wg |2(p)λ4

,a local quantity depending on the Weyl tensor.

For n ≤ 5 the correction is of global nature. Heuristics: if u ' Up,λ then

Lgu := −cn∆u+Rgu ' Un+2n−2

p,λ '1

λδp.

At large scales an approximate solution looks like the Green’s functionGp of the operator Lg. If Gp ' 1

|x|n−2 +A at p, the correction is −A/λn−2.

Andrea Malchiodi (SNS) Banff, May 7th 2019 5 / 16

Page 21: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

On the inequality Y (M, [g]) < Sn

The inequality is proved using Aubin-Talenti’s functions. Given p ∈M ,

consider a conformal metric g ' U4

n−2

p,λ g with λ large.

Since locally(M, g) ' Rn and since Up,λ is highly concentrated, QSY (Up,λ) ' Sn,with small correction terms due to the geometry of M .

Since Up,λ decays like 1|x|n−2 at infinity, it is more localized in large dimen-

sion. Aubin proved that for n ≥ 6 the corrections are given by − |Wg |2(p)λ4

,a local quantity depending on the Weyl tensor.

For n ≤ 5 the correction is of global nature. Heuristics: if u ' Up,λ then

Lgu := −cn∆u+Rgu ' Un+2n−2

p,λ '1

λδp.

At large scales an approximate solution looks like the Green’s functionGp of the operator Lg. If Gp ' 1

|x|n−2 +A at p, the correction is −A/λn−2.

Andrea Malchiodi (SNS) Banff, May 7th 2019 5 / 16

Page 22: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

On the inequality Y (M, [g]) < Sn

The inequality is proved using Aubin-Talenti’s functions. Given p ∈M ,

consider a conformal metric g ' U4

n−2

p,λ g with λ large. Since locally(M, g) ' Rn and since Up,λ is highly concentrated, QSY (Up,λ) ' Sn,with small correction terms due to the geometry of M .

Since Up,λ decays like 1|x|n−2 at infinity, it is more localized in large dimen-

sion. Aubin proved that for n ≥ 6 the corrections are given by − |Wg |2(p)λ4

,a local quantity depending on the Weyl tensor.

For n ≤ 5 the correction is of global nature. Heuristics: if u ' Up,λ then

Lgu := −cn∆u+Rgu ' Un+2n−2

p,λ '1

λδp.

At large scales an approximate solution looks like the Green’s functionGp of the operator Lg. If Gp ' 1

|x|n−2 +A at p, the correction is −A/λn−2.

Andrea Malchiodi (SNS) Banff, May 7th 2019 5 / 16

Page 23: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

On the inequality Y (M, [g]) < Sn

The inequality is proved using Aubin-Talenti’s functions. Given p ∈M ,

consider a conformal metric g ' U4

n−2

p,λ g with λ large. Since locally(M, g) ' Rn and since Up,λ is highly concentrated, QSY (Up,λ) ' Sn,with small correction terms due to the geometry of M .

Since Up,λ decays like 1|x|n−2 at infinity, it is more localized in large dimen-

sion.

Aubin proved that for n ≥ 6 the corrections are given by − |Wg |2(p)λ4

,a local quantity depending on the Weyl tensor.

For n ≤ 5 the correction is of global nature. Heuristics: if u ' Up,λ then

Lgu := −cn∆u+Rgu ' Un+2n−2

p,λ '1

λδp.

At large scales an approximate solution looks like the Green’s functionGp of the operator Lg. If Gp ' 1

|x|n−2 +A at p, the correction is −A/λn−2.

Andrea Malchiodi (SNS) Banff, May 7th 2019 5 / 16

Page 24: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

On the inequality Y (M, [g]) < Sn

The inequality is proved using Aubin-Talenti’s functions. Given p ∈M ,

consider a conformal metric g ' U4

n−2

p,λ g with λ large. Since locally(M, g) ' Rn and since Up,λ is highly concentrated, QSY (Up,λ) ' Sn,with small correction terms due to the geometry of M .

Since Up,λ decays like 1|x|n−2 at infinity, it is more localized in large dimen-

sion. Aubin proved that for n ≥ 6 the corrections are given by − |Wg |2(p)λ4

,a local quantity depending on the Weyl tensor.

For n ≤ 5 the correction is of global nature. Heuristics: if u ' Up,λ then

Lgu := −cn∆u+Rgu ' Un+2n−2

p,λ '1

λδp.

At large scales an approximate solution looks like the Green’s functionGp of the operator Lg. If Gp ' 1

|x|n−2 +A at p, the correction is −A/λn−2.

Andrea Malchiodi (SNS) Banff, May 7th 2019 5 / 16

Page 25: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

On the inequality Y (M, [g]) < Sn

The inequality is proved using Aubin-Talenti’s functions. Given p ∈M ,

consider a conformal metric g ' U4

n−2

p,λ g with λ large. Since locally(M, g) ' Rn and since Up,λ is highly concentrated, QSY (Up,λ) ' Sn,with small correction terms due to the geometry of M .

Since Up,λ decays like 1|x|n−2 at infinity, it is more localized in large dimen-

sion. Aubin proved that for n ≥ 6 the corrections are given by − |Wg |2(p)λ4

,a local quantity depending on the Weyl tensor.

For n ≤ 5 the correction is of global nature.

Heuristics: if u ' Up,λ then

Lgu := −cn∆u+Rgu ' Un+2n−2

p,λ '1

λδp.

At large scales an approximate solution looks like the Green’s functionGp of the operator Lg. If Gp ' 1

|x|n−2 +A at p, the correction is −A/λn−2.

Andrea Malchiodi (SNS) Banff, May 7th 2019 5 / 16

Page 26: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

On the inequality Y (M, [g]) < Sn

The inequality is proved using Aubin-Talenti’s functions. Given p ∈M ,

consider a conformal metric g ' U4

n−2

p,λ g with λ large. Since locally(M, g) ' Rn and since Up,λ is highly concentrated, QSY (Up,λ) ' Sn,with small correction terms due to the geometry of M .

Since Up,λ decays like 1|x|n−2 at infinity, it is more localized in large dimen-

sion. Aubin proved that for n ≥ 6 the corrections are given by − |Wg |2(p)λ4

,a local quantity depending on the Weyl tensor.

For n ≤ 5 the correction is of global nature. Heuristics:

if u ' Up,λ then

Lgu := −cn∆u+Rgu ' Un+2n−2

p,λ '1

λδp.

At large scales an approximate solution looks like the Green’s functionGp of the operator Lg. If Gp ' 1

|x|n−2 +A at p, the correction is −A/λn−2.

Andrea Malchiodi (SNS) Banff, May 7th 2019 5 / 16

Page 27: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

On the inequality Y (M, [g]) < Sn

The inequality is proved using Aubin-Talenti’s functions. Given p ∈M ,

consider a conformal metric g ' U4

n−2

p,λ g with λ large. Since locally(M, g) ' Rn and since Up,λ is highly concentrated, QSY (Up,λ) ' Sn,with small correction terms due to the geometry of M .

Since Up,λ decays like 1|x|n−2 at infinity, it is more localized in large dimen-

sion. Aubin proved that for n ≥ 6 the corrections are given by − |Wg |2(p)λ4

,a local quantity depending on the Weyl tensor.

For n ≤ 5 the correction is of global nature. Heuristics: if u ' Up,λ then

Lgu := −cn∆u+Rgu ' Un+2n−2

p,λ '1

λδp.

At large scales an approximate solution looks like the Green’s functionGp of the operator Lg. If Gp ' 1

|x|n−2 +A at p, the correction is −A/λn−2.

Andrea Malchiodi (SNS) Banff, May 7th 2019 5 / 16

Page 28: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

On the inequality Y (M, [g]) < Sn

The inequality is proved using Aubin-Talenti’s functions. Given p ∈M ,

consider a conformal metric g ' U4

n−2

p,λ g with λ large. Since locally(M, g) ' Rn and since Up,λ is highly concentrated, QSY (Up,λ) ' Sn,with small correction terms due to the geometry of M .

Since Up,λ decays like 1|x|n−2 at infinity, it is more localized in large dimen-

sion. Aubin proved that for n ≥ 6 the corrections are given by − |Wg |2(p)λ4

,a local quantity depending on the Weyl tensor.

For n ≤ 5 the correction is of global nature. Heuristics: if u ' Up,λ then

Lgu := −cn∆u+Rgu ' Un+2n−2

p,λ '1

λδp.

At large scales an approximate solution looks like the Green’s functionGp of the operator Lg.

If Gp ' 1|x|n−2 +A at p, the correction is −A/λn−2.

Andrea Malchiodi (SNS) Banff, May 7th 2019 5 / 16

Page 29: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

On the inequality Y (M, [g]) < Sn

The inequality is proved using Aubin-Talenti’s functions. Given p ∈M ,

consider a conformal metric g ' U4

n−2

p,λ g with λ large. Since locally(M, g) ' Rn and since Up,λ is highly concentrated, QSY (Up,λ) ' Sn,with small correction terms due to the geometry of M .

Since Up,λ decays like 1|x|n−2 at infinity, it is more localized in large dimen-

sion. Aubin proved that for n ≥ 6 the corrections are given by − |Wg |2(p)λ4

,a local quantity depending on the Weyl tensor.

For n ≤ 5 the correction is of global nature. Heuristics: if u ' Up,λ then

Lgu := −cn∆u+Rgu ' Un+2n−2

p,λ '1

λδp.

At large scales an approximate solution looks like the Green’s functionGp of the operator Lg. If Gp ' 1

|x|n−2 +A at p, the correction is −A/λn−2.Andrea Malchiodi (SNS) Banff, May 7th 2019 5 / 16

Page 30: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

A brief excursion in general relativity

To understand the value of A, general relativity comes into play.

A manifold (N3, g) is said to be asymptotically flat if it is a union ofa compact set K (possibly with topology), and such that N \ K isdiffeomorphic to R3 \B1(0). It is required that the metric satisfies

gij → δij at infinity (with some rate).

N

N \K

In general relativity these manifolds describe static gravitational systems.

Andrea Malchiodi (SNS) Banff, May 7th 2019 6 / 16

Page 31: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

A brief excursion in general relativity

To understand the value of A, general relativity comes into play.

A manifold (N3, g) is said to be asymptotically flat if it is a union ofa compact set K (possibly with topology), and such that N \ K isdiffeomorphic to R3 \B1(0). It is required that the metric satisfies

gij → δij at infinity (with some rate).

N

N \K

In general relativity these manifolds describe static gravitational systems.

Andrea Malchiodi (SNS) Banff, May 7th 2019 6 / 16

Page 32: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

A brief excursion in general relativity

To understand the value of A, general relativity comes into play.

A manifold (N3, g) is said to be asymptotically flat if it is a union ofa compact set K (possibly with topology), and such that N \ K isdiffeomorphic to R3 \B1(0).

It is required that the metric satisfies

gij → δij at infinity (with some rate).

N

N \K

In general relativity these manifolds describe static gravitational systems.

Andrea Malchiodi (SNS) Banff, May 7th 2019 6 / 16

Page 33: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

A brief excursion in general relativity

To understand the value of A, general relativity comes into play.

A manifold (N3, g) is said to be asymptotically flat if it is a union ofa compact set K (possibly with topology), and such that N \ K isdiffeomorphic to R3 \B1(0). It is required that the metric satisfies

gij → δij at infinity (with some rate).

N

N \K

In general relativity these manifolds describe static gravitational systems.

Andrea Malchiodi (SNS) Banff, May 7th 2019 6 / 16

Page 34: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

A brief excursion in general relativity

To understand the value of A, general relativity comes into play.

A manifold (N3, g) is said to be asymptotically flat if it is a union ofa compact set K (possibly with topology), and such that N \ K isdiffeomorphic to R3 \B1(0). It is required that the metric satisfies

gij → δij at infinity (with some rate).

N

N \K

In general relativity these manifolds describe static gravitational systems.Andrea Malchiodi (SNS) Banff, May 7th 2019 6 / 16

Page 35: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

The mass of an asymptotically flat manifold

The ADM mass m(g) ([ADM, ’60]) of such manifolds is defined as

m(g) := limr→∞

∮Sr

(∂k gjk − ∂j gkk) νjdσ.

Theorem ([Schoen-Yau, ’79])

If Rg ≥ 0 then m(g) ≥ 0. In case m(g) = 0, then (M, g) is isometric tothe flat Euclidean space (R3, dx2).

Application: Conformal blow-ups. Consider a compact Riemannianthree-manifold (M, g), and p ∈M . Define now the conformal metric

g = G4p g; Gp Green’s function of Lg with pole p.

Then (M \ p, g) is asymptotically flat, and

m(g) = limx→p

(Gp(x)− 1

d(x, p)

)= A.

Andrea Malchiodi (SNS) Banff, May 7th 2019 7 / 16

Page 36: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

The mass of an asymptotically flat manifold

The ADM mass m(g) ([ADM, ’60]) of such manifolds is defined as

m(g) := limr→∞

∮Sr

(∂k gjk − ∂j gkk) νjdσ.

Theorem ([Schoen-Yau, ’79])

If Rg ≥ 0 then m(g) ≥ 0. In case m(g) = 0, then (M, g) is isometric tothe flat Euclidean space (R3, dx2).

Application: Conformal blow-ups. Consider a compact Riemannianthree-manifold (M, g), and p ∈M . Define now the conformal metric

g = G4p g; Gp Green’s function of Lg with pole p.

Then (M \ p, g) is asymptotically flat, and

m(g) = limx→p

(Gp(x)− 1

d(x, p)

)= A.

Andrea Malchiodi (SNS) Banff, May 7th 2019 7 / 16

Page 37: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

The mass of an asymptotically flat manifold

The ADM mass m(g) ([ADM, ’60]) of such manifolds is defined as

m(g) := limr→∞

∮Sr

(∂k gjk − ∂j gkk) νjdσ.

Theorem ([Schoen-Yau, ’79])

If Rg ≥ 0 then m(g) ≥ 0. In case m(g) = 0, then (M, g) is isometric tothe flat Euclidean space (R3, dx2).

Application: Conformal blow-ups. Consider a compact Riemannianthree-manifold (M, g), and p ∈M . Define now the conformal metric

g = G4p g; Gp Green’s function of Lg with pole p.

Then (M \ p, g) is asymptotically flat, and

m(g) = limx→p

(Gp(x)− 1

d(x, p)

)= A.

Andrea Malchiodi (SNS) Banff, May 7th 2019 7 / 16

Page 38: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

The mass of an asymptotically flat manifold

The ADM mass m(g) ([ADM, ’60]) of such manifolds is defined as

m(g) := limr→∞

∮Sr

(∂k gjk − ∂j gkk) νjdσ.

Theorem ([Schoen-Yau, ’79])

If Rg ≥ 0 then m(g) ≥ 0.

In case m(g) = 0, then (M, g) is isometric tothe flat Euclidean space (R3, dx2).

Application: Conformal blow-ups. Consider a compact Riemannianthree-manifold (M, g), and p ∈M . Define now the conformal metric

g = G4p g; Gp Green’s function of Lg with pole p.

Then (M \ p, g) is asymptotically flat, and

m(g) = limx→p

(Gp(x)− 1

d(x, p)

)= A.

Andrea Malchiodi (SNS) Banff, May 7th 2019 7 / 16

Page 39: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

The mass of an asymptotically flat manifold

The ADM mass m(g) ([ADM, ’60]) of such manifolds is defined as

m(g) := limr→∞

∮Sr

(∂k gjk − ∂j gkk) νjdσ.

Theorem ([Schoen-Yau, ’79])

If Rg ≥ 0 then m(g) ≥ 0. In case m(g) = 0, then (M, g) is isometric tothe flat Euclidean space (R3, dx2).

Application: Conformal blow-ups. Consider a compact Riemannianthree-manifold (M, g), and p ∈M . Define now the conformal metric

g = G4p g; Gp Green’s function of Lg with pole p.

Then (M \ p, g) is asymptotically flat, and

m(g) = limx→p

(Gp(x)− 1

d(x, p)

)= A.

Andrea Malchiodi (SNS) Banff, May 7th 2019 7 / 16

Page 40: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

The mass of an asymptotically flat manifold

The ADM mass m(g) ([ADM, ’60]) of such manifolds is defined as

m(g) := limr→∞

∮Sr

(∂k gjk − ∂j gkk) νjdσ.

Theorem ([Schoen-Yau, ’79])

If Rg ≥ 0 then m(g) ≥ 0. In case m(g) = 0, then (M, g) is isometric tothe flat Euclidean space (R3, dx2).

Application: Conformal blow-ups.

Consider a compact Riemannianthree-manifold (M, g), and p ∈M . Define now the conformal metric

g = G4p g; Gp Green’s function of Lg with pole p.

Then (M \ p, g) is asymptotically flat, and

m(g) = limx→p

(Gp(x)− 1

d(x, p)

)= A.

Andrea Malchiodi (SNS) Banff, May 7th 2019 7 / 16

Page 41: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

The mass of an asymptotically flat manifold

The ADM mass m(g) ([ADM, ’60]) of such manifolds is defined as

m(g) := limr→∞

∮Sr

(∂k gjk − ∂j gkk) νjdσ.

Theorem ([Schoen-Yau, ’79])

If Rg ≥ 0 then m(g) ≥ 0. In case m(g) = 0, then (M, g) is isometric tothe flat Euclidean space (R3, dx2).

Application: Conformal blow-ups. Consider a compact Riemannianthree-manifold (M, g), and p ∈M .

Define now the conformal metric

g = G4p g; Gp Green’s function of Lg with pole p.

Then (M \ p, g) is asymptotically flat, and

m(g) = limx→p

(Gp(x)− 1

d(x, p)

)= A.

Andrea Malchiodi (SNS) Banff, May 7th 2019 7 / 16

Page 42: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

The mass of an asymptotically flat manifold

The ADM mass m(g) ([ADM, ’60]) of such manifolds is defined as

m(g) := limr→∞

∮Sr

(∂k gjk − ∂j gkk) νjdσ.

Theorem ([Schoen-Yau, ’79])

If Rg ≥ 0 then m(g) ≥ 0. In case m(g) = 0, then (M, g) is isometric tothe flat Euclidean space (R3, dx2).

Application: Conformal blow-ups. Consider a compact Riemannianthree-manifold (M, g), and p ∈M . Define now the conformal metric

g = G4p g; Gp Green’s function of Lg with pole p.

Then (M \ p, g) is asymptotically flat, and

m(g) = limx→p

(Gp(x)− 1

d(x, p)

)= A.

Andrea Malchiodi (SNS) Banff, May 7th 2019 7 / 16

Page 43: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

The mass of an asymptotically flat manifold

The ADM mass m(g) ([ADM, ’60]) of such manifolds is defined as

m(g) := limr→∞

∮Sr

(∂k gjk − ∂j gkk) νjdσ.

Theorem ([Schoen-Yau, ’79])

If Rg ≥ 0 then m(g) ≥ 0. In case m(g) = 0, then (M, g) is isometric tothe flat Euclidean space (R3, dx2).

Application: Conformal blow-ups. Consider a compact Riemannianthree-manifold (M, g), and p ∈M . Define now the conformal metric

g = G4p g; Gp Green’s function of Lg with pole p.

Then (M \ p, g) is asymptotically flat, and

m(g) = limx→p

(Gp(x)− 1

d(x, p)

)= A.

Andrea Malchiodi (SNS) Banff, May 7th 2019 7 / 16

Page 44: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

CR manifolds (three dimensions)

We deal with 3-D manifolds with a non-integrable two-dimensional di-stribution (contact structure) ξ, annihilated by a contact 1-form θ.

We also have a CR structure (complex rotation) J : ξ → ξ s.t. J2 = −1.

Given J as above, we have locally a vector field Z1 such that

JZ1 = iZ1; JZ1 = −iZ1 where Z1 = (Z1).

Example 1: Heisenberg group H1 = (z, t) ∈ C× R. SettingZ1=

1√2

(∂

∂z+ iz

∂t

);

Z1=

1√2

(∂

∂z− iz ∂

∂t

).

Example 2: boundaries of complex domains. Consider Ω ⊂ C2

and J2 the standard complex rotation in C2. Given p ∈ ∂Ω one canconsider the subset ξp of Tp∂Ω which is invariant by J2. We take ξp ascontact distribution, and J |ξp as the CR structure J .

Andrea Malchiodi (SNS) Banff, May 7th 2019 8 / 16

Page 45: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

CR manifolds (three dimensions)

We deal with 3-D manifolds with a non-integrable two-dimensional di-stribution (contact structure) ξ, annihilated by a contact 1-form θ.

We also have a CR structure (complex rotation) J : ξ → ξ s.t. J2 = −1.

Given J as above, we have locally a vector field Z1 such that

JZ1 = iZ1; JZ1 = −iZ1 where Z1 = (Z1).

Example 1: Heisenberg group H1 = (z, t) ∈ C× R. SettingZ1=

1√2

(∂

∂z+ iz

∂t

);

Z1=

1√2

(∂

∂z− iz ∂

∂t

).

Example 2: boundaries of complex domains. Consider Ω ⊂ C2

and J2 the standard complex rotation in C2. Given p ∈ ∂Ω one canconsider the subset ξp of Tp∂Ω which is invariant by J2. We take ξp ascontact distribution, and J |ξp as the CR structure J .

Andrea Malchiodi (SNS) Banff, May 7th 2019 8 / 16

Page 46: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

CR manifolds (three dimensions)

We deal with 3-D manifolds with a non-integrable two-dimensional di-stribution (contact structure) ξ, annihilated by a contact 1-form θ.

We also have a CR structure (complex rotation) J : ξ → ξ s.t. J2 = −1.

Given J as above, we have locally a vector field Z1 such that

JZ1 = iZ1; JZ1 = −iZ1 where Z1 = (Z1).

Example 1: Heisenberg group H1 = (z, t) ∈ C× R. SettingZ1=

1√2

(∂

∂z+ iz

∂t

);

Z1=

1√2

(∂

∂z− iz ∂

∂t

).

Example 2: boundaries of complex domains. Consider Ω ⊂ C2

and J2 the standard complex rotation in C2. Given p ∈ ∂Ω one canconsider the subset ξp of Tp∂Ω which is invariant by J2. We take ξp ascontact distribution, and J |ξp as the CR structure J .

Andrea Malchiodi (SNS) Banff, May 7th 2019 8 / 16

Page 47: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

CR manifolds (three dimensions)

We deal with 3-D manifolds with a non-integrable two-dimensional di-stribution (contact structure) ξ, annihilated by a contact 1-form θ.

We also have a CR structure (complex rotation) J : ξ → ξ s.t. J2 = −1.

Given J as above, we have locally a vector field Z1 such that

JZ1 = iZ1; JZ1 = −iZ1 where Z1 = (Z1).

Example 1: Heisenberg group H1 = (z, t) ∈ C× R. SettingZ1=

1√2

(∂

∂z+ iz

∂t

);

Z1=

1√2

(∂

∂z− iz ∂

∂t

).

Example 2: boundaries of complex domains. Consider Ω ⊂ C2

and J2 the standard complex rotation in C2. Given p ∈ ∂Ω one canconsider the subset ξp of Tp∂Ω which is invariant by J2. We take ξp ascontact distribution, and J |ξp as the CR structure J .

Andrea Malchiodi (SNS) Banff, May 7th 2019 8 / 16

Page 48: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

CR manifolds (three dimensions)

We deal with 3-D manifolds with a non-integrable two-dimensional di-stribution (contact structure) ξ, annihilated by a contact 1-form θ.

We also have a CR structure (complex rotation) J : ξ → ξ s.t. J2 = −1.

Given J as above, we have locally a vector field Z1 such that

JZ1 = iZ1; JZ1 = −iZ1 where Z1 = (Z1).

Example 1: Heisenberg group H1 = (z, t) ∈ C× R.

Setting

Z1=

1√2

(∂

∂z+ iz

∂t

);

Z1=

1√2

(∂

∂z− iz ∂

∂t

).

Example 2: boundaries of complex domains. Consider Ω ⊂ C2

and J2 the standard complex rotation in C2. Given p ∈ ∂Ω one canconsider the subset ξp of Tp∂Ω which is invariant by J2. We take ξp ascontact distribution, and J |ξp as the CR structure J .

Andrea Malchiodi (SNS) Banff, May 7th 2019 8 / 16

Page 49: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

CR manifolds (three dimensions)

We deal with 3-D manifolds with a non-integrable two-dimensional di-stribution (contact structure) ξ, annihilated by a contact 1-form θ.

We also have a CR structure (complex rotation) J : ξ → ξ s.t. J2 = −1.

Given J as above, we have locally a vector field Z1 such that

JZ1 = iZ1; JZ1 = −iZ1 where Z1 = (Z1).

Example 1: Heisenberg group H1 = (z, t) ∈ C× R. SettingZ1=

1√2

(∂

∂z+ iz

∂t

);

Z1=

1√2

(∂

∂z− iz ∂

∂t

).

Example 2: boundaries of complex domains. Consider Ω ⊂ C2

and J2 the standard complex rotation in C2. Given p ∈ ∂Ω one canconsider the subset ξp of Tp∂Ω which is invariant by J2. We take ξp ascontact distribution, and J |ξp as the CR structure J .

Andrea Malchiodi (SNS) Banff, May 7th 2019 8 / 16

Page 50: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

CR manifolds (three dimensions)

We deal with 3-D manifolds with a non-integrable two-dimensional di-stribution (contact structure) ξ, annihilated by a contact 1-form θ.

We also have a CR structure (complex rotation) J : ξ → ξ s.t. J2 = −1.

Given J as above, we have locally a vector field Z1 such that

JZ1 = iZ1; JZ1 = −iZ1 where Z1 = (Z1).

Example 1: Heisenberg group H1 = (z, t) ∈ C× R. SettingZ1=

1√2

(∂

∂z+ iz

∂t

);

Z1=

1√2

(∂

∂z− iz ∂

∂t

).

Example 2: boundaries of complex domains.

Consider Ω ⊂ C2

and J2 the standard complex rotation in C2. Given p ∈ ∂Ω one canconsider the subset ξp of Tp∂Ω which is invariant by J2. We take ξp ascontact distribution, and J |ξp as the CR structure J .

Andrea Malchiodi (SNS) Banff, May 7th 2019 8 / 16

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CR manifolds (three dimensions)

We deal with 3-D manifolds with a non-integrable two-dimensional di-stribution (contact structure) ξ, annihilated by a contact 1-form θ.

We also have a CR structure (complex rotation) J : ξ → ξ s.t. J2 = −1.

Given J as above, we have locally a vector field Z1 such that

JZ1 = iZ1; JZ1 = −iZ1 where Z1 = (Z1).

Example 1: Heisenberg group H1 = (z, t) ∈ C× R. SettingZ1=

1√2

(∂

∂z+ iz

∂t

);

Z1=

1√2

(∂

∂z− iz ∂

∂t

).

Example 2: boundaries of complex domains. Consider Ω ⊂ C2

and J2 the standard complex rotation in C2.

Given p ∈ ∂Ω one canconsider the subset ξp of Tp∂Ω which is invariant by J2. We take ξp ascontact distribution, and J |ξp as the CR structure J .

Andrea Malchiodi (SNS) Banff, May 7th 2019 8 / 16

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CR manifolds (three dimensions)

We deal with 3-D manifolds with a non-integrable two-dimensional di-stribution (contact structure) ξ, annihilated by a contact 1-form θ.

We also have a CR structure (complex rotation) J : ξ → ξ s.t. J2 = −1.

Given J as above, we have locally a vector field Z1 such that

JZ1 = iZ1; JZ1 = −iZ1 where Z1 = (Z1).

Example 1: Heisenberg group H1 = (z, t) ∈ C× R. SettingZ1=

1√2

(∂

∂z+ iz

∂t

);

Z1=

1√2

(∂

∂z− iz ∂

∂t

).

Example 2: boundaries of complex domains. Consider Ω ⊂ C2

and J2 the standard complex rotation in C2. Given p ∈ ∂Ω one canconsider the subset ξp of Tp∂Ω which is invariant by J2.

We take ξp ascontact distribution, and J |ξp as the CR structure J .

Andrea Malchiodi (SNS) Banff, May 7th 2019 8 / 16

Page 53: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

CR manifolds (three dimensions)

We deal with 3-D manifolds with a non-integrable two-dimensional di-stribution (contact structure) ξ, annihilated by a contact 1-form θ.

We also have a CR structure (complex rotation) J : ξ → ξ s.t. J2 = −1.

Given J as above, we have locally a vector field Z1 such that

JZ1 = iZ1; JZ1 = −iZ1 where Z1 = (Z1).

Example 1: Heisenberg group H1 = (z, t) ∈ C× R. SettingZ1=

1√2

(∂

∂z+ iz

∂t

);

Z1=

1√2

(∂

∂z− iz ∂

∂t

).

Example 2: boundaries of complex domains. Consider Ω ⊂ C2

and J2 the standard complex rotation in C2. Given p ∈ ∂Ω one canconsider the subset ξp of Tp∂Ω which is invariant by J2. We take ξp ascontact distribution, and J |ξp as the CR structure J .

Andrea Malchiodi (SNS) Banff, May 7th 2019 8 / 16

Page 54: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

The Webster curvature of a CR three-manifold

In 1983 Webster introduced scalar curvature W , to study the biholomor-phy problem, which behaves conformally like the scalar curvature.

Changing conformally the contact form, if θ = u2θ, then Wθ is given by

−4∆bu+Wθu = Wθu3.

Here ∆b is the sub-laplacian on M : roughly, the laplacian in the contactdirections (use Hörmander’s theory (commutators) to recover regularity).

As before, we can define a Sobolev-Webster quotient, a Webster class, andtry to uniformizeW as we did for the scalar curvature. For n ≥ 5 Jerisonand Lee (1989) proved the counterparts of Trudinger and Aubin’s results.Non-minimal solutions were found in [Gamara (et al.), ’01] for n = 3.

Andrea Malchiodi (SNS) Banff, May 7th 2019 9 / 16

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The Webster curvature of a CR three-manifold

In 1983 Webster introduced scalar curvature W , to study the biholomor-phy problem, which behaves conformally like the scalar curvature.

Changing conformally the contact form, if θ = u2θ, then Wθ is given by

−4∆bu+Wθu = Wθu3.

Here ∆b is the sub-laplacian on M : roughly, the laplacian in the contactdirections (use Hörmander’s theory (commutators) to recover regularity).

As before, we can define a Sobolev-Webster quotient, a Webster class, andtry to uniformizeW as we did for the scalar curvature. For n ≥ 5 Jerisonand Lee (1989) proved the counterparts of Trudinger and Aubin’s results.Non-minimal solutions were found in [Gamara (et al.), ’01] for n = 3.

Andrea Malchiodi (SNS) Banff, May 7th 2019 9 / 16

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The Webster curvature of a CR three-manifold

In 1983 Webster introduced scalar curvature W , to study the biholomor-phy problem, which behaves conformally like the scalar curvature.

Changing conformally the contact form, if θ = u2θ, then Wθ is given by

−4∆bu+Wθu = Wθu3.

Here ∆b is the sub-laplacian on M : roughly, the laplacian in the contactdirections (use Hörmander’s theory (commutators) to recover regularity).

As before, we can define a Sobolev-Webster quotient, a Webster class, andtry to uniformizeW as we did for the scalar curvature. For n ≥ 5 Jerisonand Lee (1989) proved the counterparts of Trudinger and Aubin’s results.Non-minimal solutions were found in [Gamara (et al.), ’01] for n = 3.

Andrea Malchiodi (SNS) Banff, May 7th 2019 9 / 16

Page 57: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

The Webster curvature of a CR three-manifold

In 1983 Webster introduced scalar curvature W , to study the biholomor-phy problem, which behaves conformally like the scalar curvature.

Changing conformally the contact form, if θ = u2θ, then Wθ is given by

−4∆bu+Wθu = Wθu3.

Here ∆b is the sub-laplacian on M : roughly, the laplacian in the contactdirections

(use Hörmander’s theory (commutators) to recover regularity).

As before, we can define a Sobolev-Webster quotient, a Webster class, andtry to uniformizeW as we did for the scalar curvature. For n ≥ 5 Jerisonand Lee (1989) proved the counterparts of Trudinger and Aubin’s results.Non-minimal solutions were found in [Gamara (et al.), ’01] for n = 3.

Andrea Malchiodi (SNS) Banff, May 7th 2019 9 / 16

Page 58: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

The Webster curvature of a CR three-manifold

In 1983 Webster introduced scalar curvature W , to study the biholomor-phy problem, which behaves conformally like the scalar curvature.

Changing conformally the contact form, if θ = u2θ, then Wθ is given by

−4∆bu+Wθu = Wθu3.

Here ∆b is the sub-laplacian on M : roughly, the laplacian in the contactdirections (use Hörmander’s theory (commutators) to recover regularity).

As before, we can define a Sobolev-Webster quotient, a Webster class, andtry to uniformizeW as we did for the scalar curvature. For n ≥ 5 Jerisonand Lee (1989) proved the counterparts of Trudinger and Aubin’s results.Non-minimal solutions were found in [Gamara (et al.), ’01] for n = 3.

Andrea Malchiodi (SNS) Banff, May 7th 2019 9 / 16

Page 59: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

The Webster curvature of a CR three-manifold

In 1983 Webster introduced scalar curvature W , to study the biholomor-phy problem, which behaves conformally like the scalar curvature.

Changing conformally the contact form, if θ = u2θ, then Wθ is given by

−4∆bu+Wθu = Wθu3.

Here ∆b is the sub-laplacian on M : roughly, the laplacian in the contactdirections (use Hörmander’s theory (commutators) to recover regularity).

As before, we can define a Sobolev-Webster quotient, a Webster class, andtry to uniformizeW as we did for the scalar curvature.

For n ≥ 5 Jerisonand Lee (1989) proved the counterparts of Trudinger and Aubin’s results.Non-minimal solutions were found in [Gamara (et al.), ’01] for n = 3.

Andrea Malchiodi (SNS) Banff, May 7th 2019 9 / 16

Page 60: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

The Webster curvature of a CR three-manifold

In 1983 Webster introduced scalar curvature W , to study the biholomor-phy problem, which behaves conformally like the scalar curvature.

Changing conformally the contact form, if θ = u2θ, then Wθ is given by

−4∆bu+Wθu = Wθu3.

Here ∆b is the sub-laplacian on M : roughly, the laplacian in the contactdirections (use Hörmander’s theory (commutators) to recover regularity).

As before, we can define a Sobolev-Webster quotient, a Webster class, andtry to uniformizeW as we did for the scalar curvature. For n ≥ 5 Jerisonand Lee (1989) proved the counterparts of Trudinger and Aubin’s results.

Non-minimal solutions were found in [Gamara (et al.), ’01] for n = 3.

Andrea Malchiodi (SNS) Banff, May 7th 2019 9 / 16

Page 61: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

The Webster curvature of a CR three-manifold

In 1983 Webster introduced scalar curvature W , to study the biholomor-phy problem, which behaves conformally like the scalar curvature.

Changing conformally the contact form, if θ = u2θ, then Wθ is given by

−4∆bu+Wθu = Wθu3.

Here ∆b is the sub-laplacian on M : roughly, the laplacian in the contactdirections (use Hörmander’s theory (commutators) to recover regularity).

As before, we can define a Sobolev-Webster quotient, a Webster class, andtry to uniformizeW as we did for the scalar curvature. For n ≥ 5 Jerisonand Lee (1989) proved the counterparts of Trudinger and Aubin’s results.Non-minimal solutions were found in [Gamara (et al.), ’01] for n = 3.

Andrea Malchiodi (SNS) Banff, May 7th 2019 9 / 16

Page 62: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

Green’s function and mass in three dimensions (CR)

In 3D the Green’s function still appears. In suitable coordinates at p ∈M

Gp '1

ρ2+A,

where ρ4(z, t) = |z|4 + t2, (z, t) ∈ H1 is the homogeneous distance.Blowing-up the contact form θ using Gp, we obtain an asymptotically(Heisenberg) flat manifold and define its mass, proportional to A.

However, one crucial difference between dimension three and higher isthe embeddability of abstract CR manifolds ([Chen-Shaw, ’01]). There isa fourth-order (Paneitz) operator P = ∆2

b + l.o.t. which plays a role here.

Theorem ([Chanillo-Chiu-Yang, ’12]) Let M3 be a compact CRmanifold. If P ≥ 0 and W > 0, then M embeds into some CN .

Andrea Malchiodi (SNS) Banff, May 7th 2019 10 / 16

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Green’s function and mass in three dimensions (CR)

In 3D the Green’s function still appears.

In suitable coordinates at p ∈M

Gp '1

ρ2+A,

where ρ4(z, t) = |z|4 + t2, (z, t) ∈ H1 is the homogeneous distance.Blowing-up the contact form θ using Gp, we obtain an asymptotically(Heisenberg) flat manifold and define its mass, proportional to A.

However, one crucial difference between dimension three and higher isthe embeddability of abstract CR manifolds ([Chen-Shaw, ’01]). There isa fourth-order (Paneitz) operator P = ∆2

b + l.o.t. which plays a role here.

Theorem ([Chanillo-Chiu-Yang, ’12]) Let M3 be a compact CRmanifold. If P ≥ 0 and W > 0, then M embeds into some CN .

Andrea Malchiodi (SNS) Banff, May 7th 2019 10 / 16

Page 64: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

Green’s function and mass in three dimensions (CR)

In 3D the Green’s function still appears. In suitable coordinates at p ∈M

Gp '1

ρ2+A,

where ρ4(z, t) = |z|4 + t2, (z, t) ∈ H1 is the homogeneous distance.

Blowing-up the contact form θ using Gp, we obtain an asymptotically(Heisenberg) flat manifold and define its mass, proportional to A.

However, one crucial difference between dimension three and higher isthe embeddability of abstract CR manifolds ([Chen-Shaw, ’01]). There isa fourth-order (Paneitz) operator P = ∆2

b + l.o.t. which plays a role here.

Theorem ([Chanillo-Chiu-Yang, ’12]) Let M3 be a compact CRmanifold. If P ≥ 0 and W > 0, then M embeds into some CN .

Andrea Malchiodi (SNS) Banff, May 7th 2019 10 / 16

Page 65: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

Green’s function and mass in three dimensions (CR)

In 3D the Green’s function still appears. In suitable coordinates at p ∈M

Gp '1

ρ2+A,

where ρ4(z, t) = |z|4 + t2, (z, t) ∈ H1 is the homogeneous distance.Blowing-up the contact form θ using Gp, we obtain an asymptotically(Heisenberg) flat manifold and define its mass, proportional to A.

However, one crucial difference between dimension three and higher isthe embeddability of abstract CR manifolds ([Chen-Shaw, ’01]). There isa fourth-order (Paneitz) operator P = ∆2

b + l.o.t. which plays a role here.

Theorem ([Chanillo-Chiu-Yang, ’12]) Let M3 be a compact CRmanifold. If P ≥ 0 and W > 0, then M embeds into some CN .

Andrea Malchiodi (SNS) Banff, May 7th 2019 10 / 16

Page 66: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

Green’s function and mass in three dimensions (CR)

In 3D the Green’s function still appears. In suitable coordinates at p ∈M

Gp '1

ρ2+A,

where ρ4(z, t) = |z|4 + t2, (z, t) ∈ H1 is the homogeneous distance.Blowing-up the contact form θ using Gp, we obtain an asymptotically(Heisenberg) flat manifold and define its mass, proportional to A.

However, one crucial difference between dimension three and higher isthe embeddability of abstract CR manifolds ([Chen-Shaw, ’01]).

There isa fourth-order (Paneitz) operator P = ∆2

b + l.o.t. which plays a role here.

Theorem ([Chanillo-Chiu-Yang, ’12]) Let M3 be a compact CRmanifold. If P ≥ 0 and W > 0, then M embeds into some CN .

Andrea Malchiodi (SNS) Banff, May 7th 2019 10 / 16

Page 67: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

Green’s function and mass in three dimensions (CR)

In 3D the Green’s function still appears. In suitable coordinates at p ∈M

Gp '1

ρ2+A,

where ρ4(z, t) = |z|4 + t2, (z, t) ∈ H1 is the homogeneous distance.Blowing-up the contact form θ using Gp, we obtain an asymptotically(Heisenberg) flat manifold and define its mass, proportional to A.

However, one crucial difference between dimension three and higher isthe embeddability of abstract CR manifolds ([Chen-Shaw, ’01]). There isa fourth-order (Paneitz) operator P = ∆2

b + l.o.t. which plays a role here.

Theorem ([Chanillo-Chiu-Yang, ’12]) Let M3 be a compact CRmanifold. If P ≥ 0 and W > 0, then M embeds into some CN .

Andrea Malchiodi (SNS) Banff, May 7th 2019 10 / 16

Page 68: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

Green’s function and mass in three dimensions (CR)

In 3D the Green’s function still appears. In suitable coordinates at p ∈M

Gp '1

ρ2+A,

where ρ4(z, t) = |z|4 + t2, (z, t) ∈ H1 is the homogeneous distance.Blowing-up the contact form θ using Gp, we obtain an asymptotically(Heisenberg) flat manifold and define its mass, proportional to A.

However, one crucial difference between dimension three and higher isthe embeddability of abstract CR manifolds ([Chen-Shaw, ’01]). There isa fourth-order (Paneitz) operator P = ∆2

b + l.o.t. which plays a role here.

Theorem ([Chanillo-Chiu-Yang, ’12])

Let M3 be a compact CRmanifold. If P ≥ 0 and W > 0, then M embeds into some CN .

Andrea Malchiodi (SNS) Banff, May 7th 2019 10 / 16

Page 69: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

Green’s function and mass in three dimensions (CR)

In 3D the Green’s function still appears. In suitable coordinates at p ∈M

Gp '1

ρ2+A,

where ρ4(z, t) = |z|4 + t2, (z, t) ∈ H1 is the homogeneous distance.Blowing-up the contact form θ using Gp, we obtain an asymptotically(Heisenberg) flat manifold and define its mass, proportional to A.

However, one crucial difference between dimension three and higher isthe embeddability of abstract CR manifolds ([Chen-Shaw, ’01]). There isa fourth-order (Paneitz) operator P = ∆2

b + l.o.t. which plays a role here.

Theorem ([Chanillo-Chiu-Yang, ’12]) Let M3 be a compact CRmanifold.

If P ≥ 0 and W > 0, then M embeds into some CN .

Andrea Malchiodi (SNS) Banff, May 7th 2019 10 / 16

Page 70: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

Green’s function and mass in three dimensions (CR)

In 3D the Green’s function still appears. In suitable coordinates at p ∈M

Gp '1

ρ2+A,

where ρ4(z, t) = |z|4 + t2, (z, t) ∈ H1 is the homogeneous distance.Blowing-up the contact form θ using Gp, we obtain an asymptotically(Heisenberg) flat manifold and define its mass, proportional to A.

However, one crucial difference between dimension three and higher isthe embeddability of abstract CR manifolds ([Chen-Shaw, ’01]). There isa fourth-order (Paneitz) operator P = ∆2

b + l.o.t. which plays a role here.

Theorem ([Chanillo-Chiu-Yang, ’12]) Let M3 be a compact CRmanifold. If P ≥ 0 and W > 0, then M embeds into some CN .

Andrea Malchiodi (SNS) Banff, May 7th 2019 10 / 16

Page 71: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

A positive mass theorem

Theorem 1 ([Cheng-M.-Yang, ’17])

Let (M3, J, θ) be a compact CR manifold. Suppose the Webster class ispositive, and that the Paneitz operator P is non-negative. Let p ∈ Mand let θ be a blown-up of contact form at p. Then

(a) the CR mass m of (M,J, θ) is non negative;

(b) if m = 0, (M,J, θ) is conformally equivalent to a standard S3(' H1).

• The proof uses a tricky integration by parts: the main idea was tobring-in the Paneitz operator to write the mass as sum of squares.

• Positivity of the mass implies that the Sobolev-Webster quotient of themanifold is lower than that of the sphere, and minimizers exist.

Andrea Malchiodi (SNS) Banff, May 7th 2019 11 / 16

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A positive mass theorem

Theorem 1 ([Cheng-M.-Yang, ’17])

Let (M3, J, θ) be a compact CR manifold. Suppose the Webster class ispositive, and that the Paneitz operator P is non-negative. Let p ∈ Mand let θ be a blown-up of contact form at p. Then

(a) the CR mass m of (M,J, θ) is non negative;

(b) if m = 0, (M,J, θ) is conformally equivalent to a standard S3(' H1).

• The proof uses a tricky integration by parts: the main idea was tobring-in the Paneitz operator to write the mass as sum of squares.

• Positivity of the mass implies that the Sobolev-Webster quotient of themanifold is lower than that of the sphere, and minimizers exist.

Andrea Malchiodi (SNS) Banff, May 7th 2019 11 / 16

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A positive mass theorem

Theorem 1 ([Cheng-M.-Yang, ’17])

Let (M3, J, θ) be a compact CR manifold.

Suppose the Webster class ispositive, and that the Paneitz operator P is non-negative. Let p ∈ Mand let θ be a blown-up of contact form at p. Then

(a) the CR mass m of (M,J, θ) is non negative;

(b) if m = 0, (M,J, θ) is conformally equivalent to a standard S3(' H1).

• The proof uses a tricky integration by parts: the main idea was tobring-in the Paneitz operator to write the mass as sum of squares.

• Positivity of the mass implies that the Sobolev-Webster quotient of themanifold is lower than that of the sphere, and minimizers exist.

Andrea Malchiodi (SNS) Banff, May 7th 2019 11 / 16

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A positive mass theorem

Theorem 1 ([Cheng-M.-Yang, ’17])

Let (M3, J, θ) be a compact CR manifold. Suppose the Webster class ispositive, and that the Paneitz operator P is non-negative.

Let p ∈ Mand let θ be a blown-up of contact form at p. Then

(a) the CR mass m of (M,J, θ) is non negative;

(b) if m = 0, (M,J, θ) is conformally equivalent to a standard S3(' H1).

• The proof uses a tricky integration by parts: the main idea was tobring-in the Paneitz operator to write the mass as sum of squares.

• Positivity of the mass implies that the Sobolev-Webster quotient of themanifold is lower than that of the sphere, and minimizers exist.

Andrea Malchiodi (SNS) Banff, May 7th 2019 11 / 16

Page 75: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

A positive mass theorem

Theorem 1 ([Cheng-M.-Yang, ’17])

Let (M3, J, θ) be a compact CR manifold. Suppose the Webster class ispositive, and that the Paneitz operator P is non-negative. Let p ∈ Mand let θ be a blown-up of contact form at p.

Then

(a) the CR mass m of (M,J, θ) is non negative;

(b) if m = 0, (M,J, θ) is conformally equivalent to a standard S3(' H1).

• The proof uses a tricky integration by parts: the main idea was tobring-in the Paneitz operator to write the mass as sum of squares.

• Positivity of the mass implies that the Sobolev-Webster quotient of themanifold is lower than that of the sphere, and minimizers exist.

Andrea Malchiodi (SNS) Banff, May 7th 2019 11 / 16

Page 76: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

A positive mass theorem

Theorem 1 ([Cheng-M.-Yang, ’17])

Let (M3, J, θ) be a compact CR manifold. Suppose the Webster class ispositive, and that the Paneitz operator P is non-negative. Let p ∈ Mand let θ be a blown-up of contact form at p. Then

(a) the CR mass m of (M,J, θ) is non negative;

(b) if m = 0, (M,J, θ) is conformally equivalent to a standard S3(' H1).

• The proof uses a tricky integration by parts: the main idea was tobring-in the Paneitz operator to write the mass as sum of squares.

• Positivity of the mass implies that the Sobolev-Webster quotient of themanifold is lower than that of the sphere, and minimizers exist.

Andrea Malchiodi (SNS) Banff, May 7th 2019 11 / 16

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A positive mass theorem

Theorem 1 ([Cheng-M.-Yang, ’17])

Let (M3, J, θ) be a compact CR manifold. Suppose the Webster class ispositive, and that the Paneitz operator P is non-negative. Let p ∈ Mand let θ be a blown-up of contact form at p. Then

(a) the CR mass m of (M,J, θ) is non negative;

(b) if m = 0, (M,J, θ) is conformally equivalent to a standard S3(' H1).

• The proof uses a tricky integration by parts: the main idea was tobring-in the Paneitz operator to write the mass as sum of squares.

• Positivity of the mass implies that the Sobolev-Webster quotient of themanifold is lower than that of the sphere, and minimizers exist.

Andrea Malchiodi (SNS) Banff, May 7th 2019 11 / 16

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A positive mass theorem

Theorem 1 ([Cheng-M.-Yang, ’17])

Let (M3, J, θ) be a compact CR manifold. Suppose the Webster class ispositive, and that the Paneitz operator P is non-negative. Let p ∈ Mand let θ be a blown-up of contact form at p. Then

(a) the CR mass m of (M,J, θ) is non negative;

(b) if m = 0, (M,J, θ) is conformally equivalent to a standard S3(' H1).

• The proof uses a tricky integration by parts

: the main idea was tobring-in the Paneitz operator to write the mass as sum of squares.

• Positivity of the mass implies that the Sobolev-Webster quotient of themanifold is lower than that of the sphere, and minimizers exist.

Andrea Malchiodi (SNS) Banff, May 7th 2019 11 / 16

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A positive mass theorem

Theorem 1 ([Cheng-M.-Yang, ’17])

Let (M3, J, θ) be a compact CR manifold. Suppose the Webster class ispositive, and that the Paneitz operator P is non-negative. Let p ∈ Mand let θ be a blown-up of contact form at p. Then

(a) the CR mass m of (M,J, θ) is non negative;

(b) if m = 0, (M,J, θ) is conformally equivalent to a standard S3(' H1).

• The proof uses a tricky integration by parts: the main idea was tobring-in the Paneitz operator to write the mass as sum of squares.

• Positivity of the mass implies that the Sobolev-Webster quotient of themanifold is lower than that of the sphere, and minimizers exist.

Andrea Malchiodi (SNS) Banff, May 7th 2019 11 / 16

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A positive mass theorem

Theorem 1 ([Cheng-M.-Yang, ’17])

Let (M3, J, θ) be a compact CR manifold. Suppose the Webster class ispositive, and that the Paneitz operator P is non-negative. Let p ∈ Mand let θ be a blown-up of contact form at p. Then

(a) the CR mass m of (M,J, θ) is non negative;

(b) if m = 0, (M,J, θ) is conformally equivalent to a standard S3(' H1).

• The proof uses a tricky integration by parts: the main idea was tobring-in the Paneitz operator to write the mass as sum of squares.

• Positivity of the mass implies that the Sobolev-Webster quotient of themanifold is lower than that of the sphere, and minimizers exist.

Andrea Malchiodi (SNS) Banff, May 7th 2019 11 / 16

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On the positivity condition for the Paneitz operator

Consider S3 in C2. Its standard CR structure J(0) is given by

J(0)ZS3

1 = iZS3

1 ; ZS3

1 = z2∂

∂z1− z1 ∂

∂z2.

It turns out that most perturbations of the standard structure are nonembeddable ([Burns-Epstein, ’90]).

Interesting case are Rossi spheres S3s , from [H.Rossi, ’65]: these are

homogeneous, of positive Webster class, have the same contact structureas the standard S3 but a distorted complex rotation J(s) for s ∈ (−ε, ε)

J(s)(ZS3

1 + sZS3

1 ) = i(ZS

3

1 + sZS3

1

).

In these cases the Paneitz operator cannot be positive-definite.

Theorem 2 ([Cheng-M.-Yang, ’19])

For small s 6= 0, the CR mass of S3s is negative (ms ' −18πs2).

Andrea Malchiodi (SNS) Banff, May 7th 2019 12 / 16

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On the positivity condition for the Paneitz operator

Consider S3 in C2. Its standard CR structure J(0) is given by

J(0)ZS3

1 = iZS3

1 ; ZS3

1 = z2∂

∂z1− z1 ∂

∂z2.

It turns out that most perturbations of the standard structure are nonembeddable ([Burns-Epstein, ’90]).

Interesting case are Rossi spheres S3s , from [H.Rossi, ’65]: these are

homogeneous, of positive Webster class, have the same contact structureas the standard S3 but a distorted complex rotation J(s) for s ∈ (−ε, ε)

J(s)(ZS3

1 + sZS3

1 ) = i(ZS

3

1 + sZS3

1

).

In these cases the Paneitz operator cannot be positive-definite.

Theorem 2 ([Cheng-M.-Yang, ’19])

For small s 6= 0, the CR mass of S3s is negative (ms ' −18πs2).

Andrea Malchiodi (SNS) Banff, May 7th 2019 12 / 16

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On the positivity condition for the Paneitz operator

Consider S3 in C2. Its standard CR structure J(0) is given by

J(0)ZS3

1 = iZS3

1 ; ZS3

1 = z2∂

∂z1− z1 ∂

∂z2.

It turns out that most perturbations of the standard structure are nonembeddable ([Burns-Epstein, ’90]).

Interesting case are Rossi spheres S3s , from [H.Rossi, ’65]: these are

homogeneous, of positive Webster class, have the same contact structureas the standard S3 but a distorted complex rotation J(s) for s ∈ (−ε, ε)

J(s)(ZS3

1 + sZS3

1 ) = i(ZS

3

1 + sZS3

1

).

In these cases the Paneitz operator cannot be positive-definite.

Theorem 2 ([Cheng-M.-Yang, ’19])

For small s 6= 0, the CR mass of S3s is negative (ms ' −18πs2).

Andrea Malchiodi (SNS) Banff, May 7th 2019 12 / 16

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On the positivity condition for the Paneitz operator

Consider S3 in C2. Its standard CR structure J(0) is given by

J(0)ZS3

1 = iZS3

1 ; ZS3

1 = z2∂

∂z1− z1 ∂

∂z2.

It turns out that most perturbations of the standard structure are nonembeddable ([Burns-Epstein, ’90]).

Interesting case are Rossi spheres S3s , from [H.Rossi, ’65]

: these arehomogeneous, of positive Webster class, have the same contact structureas the standard S3 but a distorted complex rotation J(s) for s ∈ (−ε, ε)

J(s)(ZS3

1 + sZS3

1 ) = i(ZS

3

1 + sZS3

1

).

In these cases the Paneitz operator cannot be positive-definite.

Theorem 2 ([Cheng-M.-Yang, ’19])

For small s 6= 0, the CR mass of S3s is negative (ms ' −18πs2).

Andrea Malchiodi (SNS) Banff, May 7th 2019 12 / 16

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On the positivity condition for the Paneitz operator

Consider S3 in C2. Its standard CR structure J(0) is given by

J(0)ZS3

1 = iZS3

1 ; ZS3

1 = z2∂

∂z1− z1 ∂

∂z2.

It turns out that most perturbations of the standard structure are nonembeddable ([Burns-Epstein, ’90]).

Interesting case are Rossi spheres S3s , from [H.Rossi, ’65]: these are

homogeneous, of positive Webster class, have the same contact structureas the standard S3 but a distorted complex rotation J(s) for s ∈ (−ε, ε)

J(s)(ZS3

1 + sZS3

1 ) = i(ZS

3

1 + sZS3

1

).

In these cases the Paneitz operator cannot be positive-definite.

Theorem 2 ([Cheng-M.-Yang, ’19])

For small s 6= 0, the CR mass of S3s is negative (ms ' −18πs2).

Andrea Malchiodi (SNS) Banff, May 7th 2019 12 / 16

Page 86: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

On the positivity condition for the Paneitz operator

Consider S3 in C2. Its standard CR structure J(0) is given by

J(0)ZS3

1 = iZS3

1 ; ZS3

1 = z2∂

∂z1− z1 ∂

∂z2.

It turns out that most perturbations of the standard structure are nonembeddable ([Burns-Epstein, ’90]).

Interesting case are Rossi spheres S3s , from [H.Rossi, ’65]: these are

homogeneous, of positive Webster class, have the same contact structureas the standard S3 but a distorted complex rotation J(s) for s ∈ (−ε, ε)

J(s)(ZS3

1 + sZS3

1 ) = i(ZS

3

1 + sZS3

1

).

In these cases the Paneitz operator cannot be positive-definite.

Theorem 2 ([Cheng-M.-Yang, ’19])

For small s 6= 0, the CR mass of S3s is negative (ms ' −18πs2).

Andrea Malchiodi (SNS) Banff, May 7th 2019 12 / 16

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On the positivity condition for the Paneitz operator

Consider S3 in C2. Its standard CR structure J(0) is given by

J(0)ZS3

1 = iZS3

1 ; ZS3

1 = z2∂

∂z1− z1 ∂

∂z2.

It turns out that most perturbations of the standard structure are nonembeddable ([Burns-Epstein, ’90]).

Interesting case are Rossi spheres S3s , from [H.Rossi, ’65]: these are

homogeneous, of positive Webster class, have the same contact structureas the standard S3 but a distorted complex rotation J(s) for s ∈ (−ε, ε)

J(s)(ZS3

1 + sZS3

1 ) = i(ZS

3

1 + sZS3

1

).

In these cases the Paneitz operator cannot be positive-definite.

Theorem 2 ([Cheng-M.-Yang, ’19])

For small s 6= 0, the CR mass of S3s is negative (ms ' −18πs2).

Andrea Malchiodi (SNS) Banff, May 7th 2019 12 / 16

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On the positivity condition for the Paneitz operator

Consider S3 in C2. Its standard CR structure J(0) is given by

J(0)ZS3

1 = iZS3

1 ; ZS3

1 = z2∂

∂z1− z1 ∂

∂z2.

It turns out that most perturbations of the standard structure are nonembeddable ([Burns-Epstein, ’90]).

Interesting case are Rossi spheres S3s , from [H.Rossi, ’65]: these are

homogeneous, of positive Webster class, have the same contact structureas the standard S3 but a distorted complex rotation J(s) for s ∈ (−ε, ε)

J(s)(ZS3

1 + sZS3

1 ) = i(ZS

3

1 + sZS3

1

).

In these cases the Paneitz operator cannot be positive-definite.

Theorem 2 ([Cheng-M.-Yang, ’19])

For small s 6= 0, the CR mass of S3s is negative (ms ' −18πs2).

Andrea Malchiodi (SNS) Banff, May 7th 2019 12 / 16

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Some ideas of the proof

As we saw, the mass is related to the next-order term in the expansionof the Green’s function (Robin’s function). Determining it is in generala hard problem, since it is a global object.

Fixing a pole p ∈ S3, we find suitable s-coordinates (near p) to expandthe Green’s function as Gp,(s) ' 1

ρ2(s)

+A(s), with A(s) unknown.

On the other hand, it is possible to Taylor-expand in s the equation

−4∆(s)b G(s) +W(s)G(s) = δp

away from p, in the standard coordinates of C2.

One then needs to verify that the two expansions match, obtaining thenthe asymptotic behaviour for s→ 0 of A(s), proportional to the mass.

Andrea Malchiodi (SNS) Banff, May 7th 2019 13 / 16

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Some ideas of the proof

As we saw, the mass is related to the next-order term in the expansionof the Green’s function (Robin’s function).

Determining it is in generala hard problem, since it is a global object.

Fixing a pole p ∈ S3, we find suitable s-coordinates (near p) to expandthe Green’s function as Gp,(s) ' 1

ρ2(s)

+A(s), with A(s) unknown.

On the other hand, it is possible to Taylor-expand in s the equation

−4∆(s)b G(s) +W(s)G(s) = δp

away from p, in the standard coordinates of C2.

One then needs to verify that the two expansions match, obtaining thenthe asymptotic behaviour for s→ 0 of A(s), proportional to the mass.

Andrea Malchiodi (SNS) Banff, May 7th 2019 13 / 16

Page 91: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

Some ideas of the proof

As we saw, the mass is related to the next-order term in the expansionof the Green’s function (Robin’s function). Determining it is in generala hard problem, since it is a global object.

Fixing a pole p ∈ S3, we find suitable s-coordinates (near p) to expandthe Green’s function as Gp,(s) ' 1

ρ2(s)

+A(s), with A(s) unknown.

On the other hand, it is possible to Taylor-expand in s the equation

−4∆(s)b G(s) +W(s)G(s) = δp

away from p, in the standard coordinates of C2.

One then needs to verify that the two expansions match, obtaining thenthe asymptotic behaviour for s→ 0 of A(s), proportional to the mass.

Andrea Malchiodi (SNS) Banff, May 7th 2019 13 / 16

Page 92: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

Some ideas of the proof

As we saw, the mass is related to the next-order term in the expansionof the Green’s function (Robin’s function). Determining it is in generala hard problem, since it is a global object.

Fixing a pole p ∈ S3, we find suitable s-coordinates (near p) to expandthe Green’s function as Gp,(s) ' 1

ρ2(s)

+A(s), with A(s) unknown.

On the other hand, it is possible to Taylor-expand in s the equation

−4∆(s)b G(s) +W(s)G(s) = δp

away from p, in the standard coordinates of C2.

One then needs to verify that the two expansions match, obtaining thenthe asymptotic behaviour for s→ 0 of A(s), proportional to the mass.

Andrea Malchiodi (SNS) Banff, May 7th 2019 13 / 16

Page 93: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

Some ideas of the proof

As we saw, the mass is related to the next-order term in the expansionof the Green’s function (Robin’s function). Determining it is in generala hard problem, since it is a global object.

Fixing a pole p ∈ S3, we find suitable s-coordinates (near p) to expandthe Green’s function as Gp,(s) ' 1

ρ2(s)

+A(s), with A(s) unknown.

On the other hand, it is possible to Taylor-expand in s the equation

−4∆(s)b G(s) +W(s)G(s) = δp

away from p, in the standard coordinates of C2.

One then needs to verify that the two expansions match, obtaining thenthe asymptotic behaviour for s→ 0 of A(s), proportional to the mass.

Andrea Malchiodi (SNS) Banff, May 7th 2019 13 / 16

Page 94: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

Some ideas of the proof

As we saw, the mass is related to the next-order term in the expansionof the Green’s function (Robin’s function). Determining it is in generala hard problem, since it is a global object.

Fixing a pole p ∈ S3, we find suitable s-coordinates (near p) to expandthe Green’s function as Gp,(s) ' 1

ρ2(s)

+A(s), with A(s) unknown.

On the other hand, it is possible to Taylor-expand in s the equation

−4∆(s)b G(s) +W(s)G(s) = δp

away from p, in the standard coordinates of C2.

One then needs to verify that the two expansions match, obtaining thenthe asymptotic behaviour for s→ 0 of A(s), proportional to the mass.

Andrea Malchiodi (SNS) Banff, May 7th 2019 13 / 16

Page 95: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

CR Sobolev quotient of Rossi spheres

Theorem 3 ([Cheng-M.-Yang, ’19])For small s 6= 0 the infimum of the Sobolev-Webster quotient of Rossispheres is not attained (and is equal to that of the standard S3).

- If a function has low Sobolev-Webster quotient on a Rossi sphere S3s it

has low Sobolev-Webster quotient also on the standard sphere S3 = S30 .

Minima for the Webster quotient on the standard S3 were classifed in[Jerison-Lee, ’88] as (CR counterparts of) Aubin-Talenti’s functions.

- For |s| 6= 0 small, the Webster quotient of the functions UCRλ has aprofile of this kind (need to use Theorem 2 for λ large)

0

Q(s)SW (UCR

λ )

λ

Remark. The CR Sobolev quotient of S3s , a closed manifold, behaves

like that of a domain in Rn!

Andrea Malchiodi (SNS) Banff, May 7th 2019 14 / 16

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CR Sobolev quotient of Rossi spheres

Theorem 3 ([Cheng-M.-Yang, ’19])

For small s 6= 0 the infimum of the Sobolev-Webster quotient of Rossispheres is not attained (and is equal to that of the standard S3).

- If a function has low Sobolev-Webster quotient on a Rossi sphere S3s it

has low Sobolev-Webster quotient also on the standard sphere S3 = S30 .

Minima for the Webster quotient on the standard S3 were classifed in[Jerison-Lee, ’88] as (CR counterparts of) Aubin-Talenti’s functions.

- For |s| 6= 0 small, the Webster quotient of the functions UCRλ has aprofile of this kind (need to use Theorem 2 for λ large)

0

Q(s)SW (UCR

λ )

λ

Remark. The CR Sobolev quotient of S3s , a closed manifold, behaves

like that of a domain in Rn!

Andrea Malchiodi (SNS) Banff, May 7th 2019 14 / 16

Page 97: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

CR Sobolev quotient of Rossi spheres

Theorem 3 ([Cheng-M.-Yang, ’19])For small s 6= 0 the infimum of the Sobolev-Webster quotient of Rossispheres is not attained (and is equal to that of the standard S3).

- If a function has low Sobolev-Webster quotient on a Rossi sphere S3s it

has low Sobolev-Webster quotient also on the standard sphere S3 = S30 .

Minima for the Webster quotient on the standard S3 were classifed in[Jerison-Lee, ’88] as (CR counterparts of) Aubin-Talenti’s functions.

- For |s| 6= 0 small, the Webster quotient of the functions UCRλ has aprofile of this kind (need to use Theorem 2 for λ large)

0

Q(s)SW (UCR

λ )

λ

Remark. The CR Sobolev quotient of S3s , a closed manifold, behaves

like that of a domain in Rn!

Andrea Malchiodi (SNS) Banff, May 7th 2019 14 / 16

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CR Sobolev quotient of Rossi spheres

Theorem 3 ([Cheng-M.-Yang, ’19])For small s 6= 0 the infimum of the Sobolev-Webster quotient of Rossispheres is not attained (and is equal to that of the standard S3).

- If a function has low Sobolev-Webster quotient on a Rossi sphere S3s it

has low Sobolev-Webster quotient also on the standard sphere S3 = S30 .

Minima for the Webster quotient on the standard S3 were classifed in[Jerison-Lee, ’88] as (CR counterparts of) Aubin-Talenti’s functions.

- For |s| 6= 0 small, the Webster quotient of the functions UCRλ has aprofile of this kind (need to use Theorem 2 for λ large)

0

Q(s)SW (UCR

λ )

λ

Remark. The CR Sobolev quotient of S3s , a closed manifold, behaves

like that of a domain in Rn!

Andrea Malchiodi (SNS) Banff, May 7th 2019 14 / 16

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CR Sobolev quotient of Rossi spheres

Theorem 3 ([Cheng-M.-Yang, ’19])For small s 6= 0 the infimum of the Sobolev-Webster quotient of Rossispheres is not attained (and is equal to that of the standard S3).

- If a function has low Sobolev-Webster quotient on a Rossi sphere S3s it

has low Sobolev-Webster quotient also on the standard sphere S3 = S30 .

Minima for the Webster quotient on the standard S3 were classifed in[Jerison-Lee, ’88] as (CR counterparts of) Aubin-Talenti’s functions.

- For |s| 6= 0 small, the Webster quotient of the functions UCRλ has aprofile of this kind (need to use Theorem 2 for λ large)

0

Q(s)SW (UCR

λ )

λ

Remark. The CR Sobolev quotient of S3s , a closed manifold, behaves

like that of a domain in Rn!

Andrea Malchiodi (SNS) Banff, May 7th 2019 14 / 16

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CR Sobolev quotient of Rossi spheres

Theorem 3 ([Cheng-M.-Yang, ’19])For small s 6= 0 the infimum of the Sobolev-Webster quotient of Rossispheres is not attained (and is equal to that of the standard S3).

- If a function has low Sobolev-Webster quotient on a Rossi sphere S3s it

has low Sobolev-Webster quotient also on the standard sphere S3 = S30 .

Minima for the Webster quotient on the standard S3 were classifed in[Jerison-Lee, ’88] as (CR counterparts of) Aubin-Talenti’s functions.

- For |s| 6= 0 small, the Webster quotient of the functions UCRλ has aprofile of this kind (need to use Theorem 2 for λ large)

0

Q(s)SW (UCR

λ )

λ

Remark. The CR Sobolev quotient of S3s , a closed manifold, behaves

like that of a domain in Rn!

Andrea Malchiodi (SNS) Banff, May 7th 2019 14 / 16

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CR Sobolev quotient of Rossi spheres

Theorem 3 ([Cheng-M.-Yang, ’19])For small s 6= 0 the infimum of the Sobolev-Webster quotient of Rossispheres is not attained (and is equal to that of the standard S3).

- If a function has low Sobolev-Webster quotient on a Rossi sphere S3s it

has low Sobolev-Webster quotient also on the standard sphere S3 = S30 .

Minima for the Webster quotient on the standard S3 were classifed in[Jerison-Lee, ’88] as (CR counterparts of) Aubin-Talenti’s functions.

- For |s| 6= 0 small, the Webster quotient of the functions UCRλ has aprofile of this kind (need to use Theorem 2 for λ large)

0

Q(s)SW (UCR

λ )

λ

Remark. The CR Sobolev quotient of S3s , a closed manifold, behaves

like that of a domain in Rn!Andrea Malchiodi (SNS) Banff, May 7th 2019 14 / 16

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Some open problems

Another problem recently settled is compactness of solutions to Yama-be’s equation ([Druet, ’04], [Li-Zhang, ’05-’06], [Brendle-Marques, ’08],[Khuri-Marques-Schoen, ’09]). Compactness holds if and only if n ≤ 24.

Compactness for the CR case is entirely open. One reason is that profilesof blow-ups are not classified. This concerns entire positive solutions to

−∆bu = uQ+2Q−2 in Hn; Q = 2n+ 2.

Assuming finite volume, it is done in [Jerison-Lee, ’88]. However we maynot have this assumption, and moving planes do not work.

A related problem concerns the classification of

−∆bu = up in Hn; p <Q+ 2

Q− 2.

In Rn it was shown in [Gidas-Spruck, ’81] that u ≡ 0. In Hn, there arepartial results in [Birindelli-Capuzzo Dolcetta-Cutrì, 97], for p < Q

Q−2 .

Andrea Malchiodi (SNS) Banff, May 7th 2019 15 / 16

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Some open problems

Another problem recently settled is compactness of solutions to Yama-be’s equation ([Druet, ’04], [Li-Zhang, ’05-’06], [Brendle-Marques, ’08],[Khuri-Marques-Schoen, ’09]).

Compactness holds if and only if n ≤ 24.

Compactness for the CR case is entirely open. One reason is that profilesof blow-ups are not classified. This concerns entire positive solutions to

−∆bu = uQ+2Q−2 in Hn; Q = 2n+ 2.

Assuming finite volume, it is done in [Jerison-Lee, ’88]. However we maynot have this assumption, and moving planes do not work.

A related problem concerns the classification of

−∆bu = up in Hn; p <Q+ 2

Q− 2.

In Rn it was shown in [Gidas-Spruck, ’81] that u ≡ 0. In Hn, there arepartial results in [Birindelli-Capuzzo Dolcetta-Cutrì, 97], for p < Q

Q−2 .

Andrea Malchiodi (SNS) Banff, May 7th 2019 15 / 16

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Some open problems

Another problem recently settled is compactness of solutions to Yama-be’s equation ([Druet, ’04], [Li-Zhang, ’05-’06], [Brendle-Marques, ’08],[Khuri-Marques-Schoen, ’09]). Compactness holds if and only if n ≤ 24.

Compactness for the CR case is entirely open. One reason is that profilesof blow-ups are not classified. This concerns entire positive solutions to

−∆bu = uQ+2Q−2 in Hn; Q = 2n+ 2.

Assuming finite volume, it is done in [Jerison-Lee, ’88]. However we maynot have this assumption, and moving planes do not work.

A related problem concerns the classification of

−∆bu = up in Hn; p <Q+ 2

Q− 2.

In Rn it was shown in [Gidas-Spruck, ’81] that u ≡ 0. In Hn, there arepartial results in [Birindelli-Capuzzo Dolcetta-Cutrì, 97], for p < Q

Q−2 .

Andrea Malchiodi (SNS) Banff, May 7th 2019 15 / 16

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Some open problems

Another problem recently settled is compactness of solutions to Yama-be’s equation ([Druet, ’04], [Li-Zhang, ’05-’06], [Brendle-Marques, ’08],[Khuri-Marques-Schoen, ’09]). Compactness holds if and only if n ≤ 24.

Compactness for the CR case is entirely open.

One reason is that profilesof blow-ups are not classified. This concerns entire positive solutions to

−∆bu = uQ+2Q−2 in Hn; Q = 2n+ 2.

Assuming finite volume, it is done in [Jerison-Lee, ’88]. However we maynot have this assumption, and moving planes do not work.

A related problem concerns the classification of

−∆bu = up in Hn; p <Q+ 2

Q− 2.

In Rn it was shown in [Gidas-Spruck, ’81] that u ≡ 0. In Hn, there arepartial results in [Birindelli-Capuzzo Dolcetta-Cutrì, 97], for p < Q

Q−2 .

Andrea Malchiodi (SNS) Banff, May 7th 2019 15 / 16

Page 106: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

Some open problems

Another problem recently settled is compactness of solutions to Yama-be’s equation ([Druet, ’04], [Li-Zhang, ’05-’06], [Brendle-Marques, ’08],[Khuri-Marques-Schoen, ’09]). Compactness holds if and only if n ≤ 24.

Compactness for the CR case is entirely open. One reason is that profilesof blow-ups are not classified.

This concerns entire positive solutions to

−∆bu = uQ+2Q−2 in Hn; Q = 2n+ 2.

Assuming finite volume, it is done in [Jerison-Lee, ’88]. However we maynot have this assumption, and moving planes do not work.

A related problem concerns the classification of

−∆bu = up in Hn; p <Q+ 2

Q− 2.

In Rn it was shown in [Gidas-Spruck, ’81] that u ≡ 0. In Hn, there arepartial results in [Birindelli-Capuzzo Dolcetta-Cutrì, 97], for p < Q

Q−2 .

Andrea Malchiodi (SNS) Banff, May 7th 2019 15 / 16

Page 107: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

Some open problems

Another problem recently settled is compactness of solutions to Yama-be’s equation ([Druet, ’04], [Li-Zhang, ’05-’06], [Brendle-Marques, ’08],[Khuri-Marques-Schoen, ’09]). Compactness holds if and only if n ≤ 24.

Compactness for the CR case is entirely open. One reason is that profilesof blow-ups are not classified. This concerns entire positive solutions to

−∆bu = uQ+2Q−2 in Hn; Q = 2n+ 2.

Assuming finite volume, it is done in [Jerison-Lee, ’88]. However we maynot have this assumption, and moving planes do not work.

A related problem concerns the classification of

−∆bu = up in Hn; p <Q+ 2

Q− 2.

In Rn it was shown in [Gidas-Spruck, ’81] that u ≡ 0. In Hn, there arepartial results in [Birindelli-Capuzzo Dolcetta-Cutrì, 97], for p < Q

Q−2 .

Andrea Malchiodi (SNS) Banff, May 7th 2019 15 / 16

Page 108: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

Some open problems

Another problem recently settled is compactness of solutions to Yama-be’s equation ([Druet, ’04], [Li-Zhang, ’05-’06], [Brendle-Marques, ’08],[Khuri-Marques-Schoen, ’09]). Compactness holds if and only if n ≤ 24.

Compactness for the CR case is entirely open. One reason is that profilesof blow-ups are not classified. This concerns entire positive solutions to

−∆bu = uQ+2Q−2 in Hn; Q = 2n+ 2.

Assuming finite volume, it is done in [Jerison-Lee, ’88].

However we maynot have this assumption, and moving planes do not work.

A related problem concerns the classification of

−∆bu = up in Hn; p <Q+ 2

Q− 2.

In Rn it was shown in [Gidas-Spruck, ’81] that u ≡ 0. In Hn, there arepartial results in [Birindelli-Capuzzo Dolcetta-Cutrì, 97], for p < Q

Q−2 .

Andrea Malchiodi (SNS) Banff, May 7th 2019 15 / 16

Page 109: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

Some open problems

Another problem recently settled is compactness of solutions to Yama-be’s equation ([Druet, ’04], [Li-Zhang, ’05-’06], [Brendle-Marques, ’08],[Khuri-Marques-Schoen, ’09]). Compactness holds if and only if n ≤ 24.

Compactness for the CR case is entirely open. One reason is that profilesof blow-ups are not classified. This concerns entire positive solutions to

−∆bu = uQ+2Q−2 in Hn; Q = 2n+ 2.

Assuming finite volume, it is done in [Jerison-Lee, ’88]. However we maynot have this assumption, and moving planes do not work.

A related problem concerns the classification of

−∆bu = up in Hn; p <Q+ 2

Q− 2.

In Rn it was shown in [Gidas-Spruck, ’81] that u ≡ 0. In Hn, there arepartial results in [Birindelli-Capuzzo Dolcetta-Cutrì, 97], for p < Q

Q−2 .

Andrea Malchiodi (SNS) Banff, May 7th 2019 15 / 16

Page 110: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

Some open problems

Another problem recently settled is compactness of solutions to Yama-be’s equation ([Druet, ’04], [Li-Zhang, ’05-’06], [Brendle-Marques, ’08],[Khuri-Marques-Schoen, ’09]). Compactness holds if and only if n ≤ 24.

Compactness for the CR case is entirely open. One reason is that profilesof blow-ups are not classified. This concerns entire positive solutions to

−∆bu = uQ+2Q−2 in Hn; Q = 2n+ 2.

Assuming finite volume, it is done in [Jerison-Lee, ’88]. However we maynot have this assumption, and moving planes do not work.

A related problem concerns the classification of

−∆bu = up in Hn; p <Q+ 2

Q− 2.

In Rn it was shown in [Gidas-Spruck, ’81] that u ≡ 0. In Hn, there arepartial results in [Birindelli-Capuzzo Dolcetta-Cutrì, 97], for p < Q

Q−2 .

Andrea Malchiodi (SNS) Banff, May 7th 2019 15 / 16

Page 111: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

Some open problems

Another problem recently settled is compactness of solutions to Yama-be’s equation ([Druet, ’04], [Li-Zhang, ’05-’06], [Brendle-Marques, ’08],[Khuri-Marques-Schoen, ’09]). Compactness holds if and only if n ≤ 24.

Compactness for the CR case is entirely open. One reason is that profilesof blow-ups are not classified. This concerns entire positive solutions to

−∆bu = uQ+2Q−2 in Hn; Q = 2n+ 2.

Assuming finite volume, it is done in [Jerison-Lee, ’88]. However we maynot have this assumption, and moving planes do not work.

A related problem concerns the classification of

−∆bu = up in Hn; p <Q+ 2

Q− 2.

In Rn it was shown in [Gidas-Spruck, ’81] that u ≡ 0.

In Hn, there arepartial results in [Birindelli-Capuzzo Dolcetta-Cutrì, 97], for p < Q

Q−2 .

Andrea Malchiodi (SNS) Banff, May 7th 2019 15 / 16

Page 112: On the Sobolev quotient in CR geometry 0.3cm Joint work ... · OntheSobolevquotientinCRgeometry Joint work with J.H.Cheng and P.Yang AndreaMalchiodi(SNS) Banff,May7th2019 AndreaMalchiodi(SNS)

Some open problems

Another problem recently settled is compactness of solutions to Yama-be’s equation ([Druet, ’04], [Li-Zhang, ’05-’06], [Brendle-Marques, ’08],[Khuri-Marques-Schoen, ’09]). Compactness holds if and only if n ≤ 24.

Compactness for the CR case is entirely open. One reason is that profilesof blow-ups are not classified. This concerns entire positive solutions to

−∆bu = uQ+2Q−2 in Hn; Q = 2n+ 2.

Assuming finite volume, it is done in [Jerison-Lee, ’88]. However we maynot have this assumption, and moving planes do not work.

A related problem concerns the classification of

−∆bu = up in Hn; p <Q+ 2

Q− 2.

In Rn it was shown in [Gidas-Spruck, ’81] that u ≡ 0. In Hn, there arepartial results in [Birindelli-Capuzzo Dolcetta-Cutrì, 97], for p < Q

Q−2 .Andrea Malchiodi (SNS) Banff, May 7th 2019 15 / 16

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Thanks for your attention

Andrea Malchiodi (SNS) Banff, May 7th 2019 16 / 16