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

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

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) =

(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).

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)

(Y ) −cn∆u+Rg = Run+2n−2 ; cn = 4

n− 1

n− 2, R ∈ R.

QSY (u) =

(cn|∇u|2 +Rgu

2)dV(∫

M |u|2∗dV

) 22∗

; 2∗ =2n

n− 2.

Y (M, [g]) = infu6≡0

QSY (u).

The Yamabe problem

(Y ) −cn∆u+Rg = Run+2n−2 ; cn = 4

n− 1

n− 2, R ∈ R.

QSY (u) =

(cn|∇u|2 +Rgu

2)dV(∫

M |u|2∗dV

) 22∗

; 2∗ =2n

n− 2.

Y (M, [g]) = infu6≡0

QSY (u).

The Yamabe problem

(Y ) −cn∆u+Rg = Run+2n−2 ; cn = 4

n− 1

n− 2, R ∈ R.

QSY (u) =

(cn|∇u|2 +Rgu

2)dV(∫

M |u|2∗dV

) 22∗

; 2∗ =2n

n− 2.

Y (M, [g]) = infu6≡0

QSY (u).

The Yamabe problem

(Y ) −cn∆u+Rg = Run+2n−2 ; cn = 4

n− 1

n− 2, R ∈ R.

QSY (u) =

(cn|∇u|2 +Rgu

2)dV(∫

M |u|2∗dV

) 22∗

; 2∗ =2n

n− 2.

Y (M, [g]) = infu6≡0

QSY (u).

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|

‖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.

‖u‖2L2∗ (Rn)

≤ Bn∫Rn

|∇u|2dx; u ∈ C∞c (Rn).

∫Rn cn|∇u|

‖u‖2L2∗ (Rn)

Up,λ(x) :=λ

n−22

(1 + λ2|x− p|2)n−22

; p ∈ Rn, λ > 0.

‖u‖2L2∗ (Rn)

≤ Bn∫Rn

|∇u|2dx; u ∈ C∞c (Rn).

∫Rn cn|∇u|

‖u‖2L2∗ (Rn)

Up,λ(x) :=λ

n−22

(1 + λ2|x− p|2)n−22

; p ∈ Rn, λ > 0.

‖u‖2L2∗ (Rn)

≤ Bn∫Rn

|∇u|2dx; u ∈ C∞c (Rn).

∫Rn cn|∇u|

‖u‖2L2∗ (Rn)

Up,λ(x) :=λ

n−22

(1 + λ2|x− p|2)n−22

; p ∈ Rn, λ > 0.

‖u‖2L2∗ (Rn)

≤ Bn∫Rn

|∇u|2dx; u ∈ C∞c (Rn).

∫Rn cn|∇u|

‖u‖2L2∗ (Rn)

Up,λ(x) :=λ

n−22

(1 + λ2|x− p|2)n−22

; p ∈ Rn, λ > 0.

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).

- 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).

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

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.

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

Given p ∈M ,

p,λ '1

λδp.

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 .

p,λ '1

λδp.

p,λ '1

λδp.

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

p,λ '1

λδp.

p,λ '1

λδp.

For n ≤ 5 the correction is of global nature.

Heuristics: if u ' Up,λ then

p,λ '1

λδp.

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

if u ' Up,λ then

p,λ '1

λδp.

p,λ '1

λδp.

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.

p,λ '1

λδp.

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

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).

In general relativity these manifolds describe static gravitational systems.

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

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

The mass of an asymptotically flat manifold

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

m(g) := limr→∞

(∂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)

m(g) := limr→∞

m(g) = limx→p

(Gp(x)− 1

d(x, p)

m(g) := limr→∞

m(g) = limx→p

(Gp(x)− 1

d(x, p)

m(g) := limr→∞

If Rg ≥ 0 then m(g) ≥ 0.

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

m(g) = limx→p

(Gp(x)− 1

d(x, p)

m(g) := limr→∞

m(g) = limx→p

(Gp(x)− 1

d(x, p)

m(g) := limr→∞

Application: Conformal blow-ups.

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

m(g) = limx→p

(Gp(x)− 1

d(x, p)

m(g) := limr→∞

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

Define now the conformal metric

m(g) = limx→p

(Gp(x)− 1

d(x, p)

m(g) := limr→∞

m(g) = limx→p

(Gp(x)− 1

d(x, p)

m(g) := limr→∞

m(g) = limx→p

(Gp(x)− 1

d(x, p)

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=

∂z+ iz

∂z− iz ∂

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 .

∂z+ iz

∂z− iz ∂

∂z+ iz

∂z− iz ∂

∂z+ iz

∂z− iz ∂

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

Setting

∂z+ iz

∂z− iz ∂

∂z+ iz

∂z− iz ∂

∂z+ iz

∂z− iz ∂

Example 2: boundaries of complex domains.

Consider Ω ⊂ C2

∂z+ iz

∂z− iz ∂

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 .

∂z+ iz

∂z− iz ∂

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 .

∂z+ iz

∂z− iz ∂

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.

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.

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.

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

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

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 .

In 3D the Green’s function still appears.

In suitable coordinates at p ∈M

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.

ρ2+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

ρ2+A,

Theorem ([Chanillo-Chiu-Yang, ’12])

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

ρ2+A,

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

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

ρ2+A,

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.

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

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

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.

• 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.

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

1 + sZS3

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

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

J(0)ZS3

1 = iZS3

1 ; ZS3

1 = z2∂

∂z1− z1 ∂

∂z2.

J(s)(ZS3

1 + sZS3

1 ) = i(ZS

1 + sZS3

J(0)ZS3

1 = iZS3

1 ; ZS3

1 = z2∂

∂z1− z1 ∂

∂z2.

J(s)(ZS3

1 + sZS3

1 ) = i(ZS

1 + sZS3

J(0)ZS3

1 = iZS3

1 ; ZS3

1 = z2∂

∂z1− z1 ∂

∂z2.

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

1 + sZS3

J(0)ZS3

1 = iZS3

1 ; ZS3

1 = z2∂

∂z1− z1 ∂

∂z2.

J(s)(ZS3

1 + sZS3

1 ) = i(ZS

1 + sZS3

J(0)ZS3

1 = iZS3

1 ; ZS3

1 = z2∂

∂z1− z1 ∂

∂z2.

J(s)(ZS3

1 + sZS3

1 ) = i(ZS

1 + sZS3

J(0)ZS3

1 = iZS3

1 ; ZS3

1 = z2∂

∂z1− z1 ∂

∂z2.

J(s)(ZS3

1 + sZS3

1 ) = i(ZS

1 + sZS3

J(0)ZS3

1 = iZS3

1 ; ZS3

1 = z2∂

∂z1− z1 ∂

∂z2.

J(s)(ZS3

1 + sZS3

1 ) = i(ZS

1 + sZS3

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.

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.

ρ2(s)

−4∆(s)b G(s) +W(s)G(s) = δp

ρ2(s)

−4∆(s)b G(s) +W(s)G(s) = δp

ρ2(s)

−4∆(s)b G(s) +W(s)G(s) = δp

ρ2(s)

−4∆(s)b G(s) +W(s)G(s) = δp

ρ2(s)

−4∆(s)b G(s) +W(s)G(s) = δp

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)

Q(s)SW (UCR

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

like that of a domain in Rn!

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).

Q(s)SW (UCR

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

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 .

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.

−∆bu = uQ+2Q−2 in Hn; Q = 2n+ 2.

Q− 2.

Q−2 .

Some open problems

−∆bu = uQ+2Q−2 in Hn; Q = 2n+ 2.

Q− 2.

Q−2 .

Some open problems

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.

Q− 2.

Q−2 .

Some open problems

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.

Q− 2.

Q−2 .

Some open problems

−∆bu = uQ+2Q−2 in Hn; Q = 2n+ 2.

Q− 2.

Q−2 .

Some open problems

−∆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.

Q− 2.

Q−2 .

Some open problems

−∆bu = uQ+2Q−2 in Hn; Q = 2n+ 2.

Q− 2.

Q−2 .

Some open problems

−∆bu = uQ+2Q−2 in Hn; Q = 2n+ 2.

Q− 2.

Q−2 .

Some open problems

−∆bu = uQ+2Q−2 in Hn; Q = 2n+ 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 .

Some open problems

−∆bu = uQ+2Q−2 in Hn; Q = 2n+ 2.

Q− 2.

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

Thanks for your attention

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

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