Min-max Theory, Willmore conjecture, and Energy of links - 0.3in...

Transcript of Min-max Theory, Willmore conjecture, and Energy of links - 0.3in...

Min-max Theory, Willmore conjecture, and Energyof links

André Neves

(Joint with Fernando Marques)
Q: What is the best way of immersing a sphere in space?

A: The one that minimizes the bending energy∫Σ

H̄2 =∫

Σ

(k1 + k2

2

)2.

• bending energy is conformally invariant;

• every compact surface has∫

ΣH̄2 ≥ 4π with equality only for round

sphere.
Q: What is the best way of immersing a sphere in space?

A: The one that minimizes the bending energy∫Σ

H̄2 =∫

Σ

(k1 + k2

2

)2.

• bending energy is conformally invariant;

• every compact surface has∫

ΣH̄2 ≥ 4π with equality only for round

sphere.
What is the best way of immersing a torus in space?

Conjecture (Willmore, ’65)For every torus Σ immersed in R3∫

Σ

H̄2 ≥ 2π2.
What is the best way of immersing a torus in space?

Conjecture (Willmore, ’65)For every torus Σ immersed in R3∫

Σ

H̄2 ≥ 2π2.
Consider stereographic projection

π : S3 − {north pole} −→ R3.

• If Σ ⊂ S3 then ∫Σ

(1 + H2) =∫π(Σ)

H̄2.

• the Willmore energy of Σ ⊂ S3 is defined to be

W(Σ) =∫

Σ

(1 + H2).

Willmore Conjecture - ’65Every torus Σ immersed in S3 hasW(Σ) ≥ 2π2.
Consider stereographic projection

π : S3 − {north pole} −→ R3.

• If Σ ⊂ S3 then ∫Σ

(1 + H2) =∫π(Σ)

H̄2.

• the Willmore energy of Σ ⊂ S3 is defined to be

W(Σ) =∫

Σ

(1 + H2).

Willmore Conjecture - ’65Every torus Σ immersed in S3 hasW(Σ) ≥ 2π2.
Theorem (Marques-N., ’12)For every embedded surface Σ ⊂ S3 with genus g ≥ 1

W(Σ) ≥ 2π2

with equality if and only if Σ is conformal to S1(

1√2

)× S1

(1√2

).

Corollary AWillmore conjecture holds.

Corollary BThe only two minimal surfaces in S3 with area ≤ 2π2 are great spheres andClifford torus.
Links in space

• γ1 and γ2 two linked curves in R3.

• lk (γ1, γ2) is the linking number

lk = 1 lk = 3
Links in space

• γ1 and γ2 two linked curves in R3.

• lk (γ1, γ2) is the linking number

lk = 1 lk = 3
Links in space

• Möbius cross energy of a link (γ1, γ2) is

E(γ1, γ2) =∫

S1×S1

|γ′1(s)||γ′2(t)||γ1(s)− γ2(t)|2

ds dt .

• E(γ1, γ2) is conformally invariant and

E(γ1, γ2) ≥ 4π|lk (γ1, γ2)|
Links in space

Q: What is the best way of immersing a non-trivial link (γ1, γ2) in R3?

A: The one that minimizes the Möbius energy.

Conjecture (Freedman-He-Wang, ’94)

If lk (γ1, γ2) = ±1, then E(γ1, γ2) ≥ 2π2.

• If true, every non-trivial link has E(γ1, γ2) ≥ 2π2.
Links in space

Q: What is the best way of immersing a non-trivial link (γ1, γ2) in R3?

A: The one that minimizes the Möbius energy.

Conjecture (Freedman-He-Wang, ’94)

If lk (γ1, γ2) = ±1, then E(γ1, γ2) ≥ 2π2.

• If true, every non-trivial link has E(γ1, γ2) ≥ 2π2.
Theorem (Agol-Marques-N., ’12)

If lk (γ1, γ2) = ±1, then E(γ1, γ2) ≥ 2π2.

If equality then (γ1, γ2) is conformal to the standard Hopf link in S3

β1(t) = (cos t , sin t ,0,0) and β2(s) = (0,0, cos s, sin s).
Theorem (Agol-Marques-N., ’12)

If lk (γ1, γ2) = ±1, then E(γ1, γ2) ≥ 2π2.

If equality then (γ1, γ2) is conformal to the standard Hopf link in S3

β1(t) = (cos t , sin t ,0,0) and β2(s) = (0,0, cos s, sin s).
Partial results...• (Willmore ’71, Shioama-Takagi ’70): Conjecture is true for tubes of

constant radius around a space curve.

• (Langevin-Rosenberg ’76): If Σ ⊂ R3 is a knotted torus thenW(Σ) ≥ 8π.

• (Chen, ’83): Conjecture is true for flat tori in 3-sphere.

• (Langer-Singer ’84): Conjecture is true for tori of revolution.
Partial results...

• (Li-Yau ’82): If Σ ⊂ S3 covers a point k times thenW(Σ) ≥ 4πk . Thus

Σ not embedded =⇒ W(Σ) ≥ 8π.

• (Li-Yau ’82): Conjecture is true for a set of conformal classes whichcontains the conformal class of the Clifford torus.

• (Montiel-Ros ’86):Conjecture is true for a larger set of conformal classesthan the set given by Li-Yau.
Partial results...




Partial results...




Partial results...

(Benisom-Mutz ’91)Conjecture is true for observed toroidal vesicles in some cells.
Partial results...

• (Simon ’93): There is a torus which has least Willmore energy among alltori.

• (Ros ’99, Topping ’00): Conjecture is true for tori symmetric under theantipodal map.

• (Ros ’00): Conjecture is true for tori symmetric with respect to a point.
Partial results...



Partial results...



Other results...

Theorem (Bryant ’84)

Classified immersions of Willmore spheres.
Other results...

Theorem (Pinkall ’85)

Found many embedded Willmore tori which are not conformal to minimalsurface.
Min-max Theory

How to find critical points for area functional?

• Z2(S3) = oriented surfaces in S3.

• φ : Ik → Z2(S3) continuous, where Ik = k -cube.• [φ] = homotopy class relative to the boundary (i.e. φ|∂Ik is fixed).• L([φ]) = infψ∈[φ] maxx∈Ik area (ψ(x)).
Min-max Theory


• Z2(S3) = oriented surfaces in S3.• φ : Ik → Z2(S3) continuous, where Ik = k -cube.

• [φ] = homotopy class relative to the boundary (i.e. φ|∂Ik is fixed).• L([φ]) = infψ∈[φ] maxx∈Ik area (ψ(x)).
Min-max Theory


• Z2(S3) = oriented surfaces in S3.• φ : Ik → Z2(S3) continuous, where Ik = k -cube.• [φ] = homotopy class relative to the boundary (i.e. φ|∂Ik is fixed).

• L([φ]) = infψ∈[φ] maxx∈Ik area (ψ(x)).
Min-max Theory


• Z2(S3) = oriented surfaces in S3.• φ : Ik → Z2(S3) continuous, where Ik = k -cube.• [φ] = homotopy class relative to the boundary (i.e. φ|∂Ik is fixed).• L([φ]) = infψ∈[φ] maxx∈Ik area (ψ(x)).
Min-max Theory

Theorem (Pitts, ’81)Assume

L([φ]) = infψ∈[φ]

maxx∈Ik

area (ψ(x)) > maxx∈∂Ik

area (φ(x)),

(this means [φ] 6= 0 ∈ πk (Z2(S3), φ|∂Ik ).)

There is Σ ⊂ S3 smooth embedded minimal surface (with multiplicities) so that

L([φ]) = area(Σ).

• Conjecture: index(Σ) ≤ k .
Min-max Theory

Theorem (Pitts, ’81)Assume

L([φ]) = infψ∈[φ]

maxx∈Ik

area (ψ(x)) > maxx∈∂Ik

area (φ(x)),

(this means [φ] 6= 0 ∈ πk (Z2(S3), φ|∂Ik ).)

There is Σ ⊂ S3 smooth embedded minimal surface (with multiplicities) so that

L([φ]) = area(Σ).

• Conjecture: index(Σ) ≤ k .
Min-max Theory

Theorem (Urbano, ’90)Σ ⊂ S3 compact embedded minimal surface with index(Σ) ≤ 5. Then either• Σ is a great sphere (index(Σ) = 1 and area(Σ) = 4π), or• Σ is the Clifford torus (index(Σ) = 5 and area(Σ) = 2π2).

(Almgren, ’62) The sweepout

φ0(t) = {x4 = 2t − 1} ∩ S3, 0 ≤ t ≤ 1,

has L([φ0]) > 0 and so L([φ0]) = 4π.
Min-max Theory

Theorem (Urbano, ’90)Σ ⊂ S3 compact embedded minimal surface with index(Σ) ≤ 5. Then either• Σ is a great sphere (index(Σ) = 1 and area(Σ) = 4π), or• Σ is the Clifford torus (index(Σ) = 5 and area(Σ) = 2π2).

(Almgren, ’62) The sweepout

φ0(t) = {x4 = 2t − 1} ∩ S3, 0 ≤ t ≤ 1,

has L([φ0]) > 0 and so L([φ0]) = 4π.
Min-max Theory

• T = {oriented great spheres} ≈ S3

• R = {oriented round spheres} ≈ S3 × [0, π]

Theorem (Marques-N.)Consider φ : I5 → Z2(S3) continuous with

1 φ(I4 × {0}) = φ(I4 × {1}) = 0;2 φ(∂I4 × I) ⊂ R and for x ∈ ∂I4, φ(x , t) ∈ T ⇐⇒ t = 1/2;3 the sweepout t 7→ φ(1/2,1/2,1/2,1/2,1/2, t) is non-trivial.

If φ : ∂I4 × {1/2} ≈ S3 → T ≈ S3 has non-zero degree then

L([φ]) > 4π = maxx∈∂I5

area(φ(x)).
Min-max Theory• T = {oriented great spheres} ≈ S3





L([φ]) > 4π = maxx∈∂I5

area(φ(x)).



1 φ(I4 × {0}) = φ(I4 × {1}) = 0;

2 φ(∂I4 × I) ⊂ R and for x ∈ ∂I4, φ(x , t) ∈ T ⇐⇒ t = 1/2;3 the sweepout t 7→ φ(1/2,1/2,1/2,1/2,1/2, t) is non-trivial.


L([φ]) > 4π = maxx∈∂I5

area(φ(x)).



1 φ(I4 × {0}) = φ(I4 × {1}) = 0;2 φ(∂I4 × I) ⊂ R and for x ∈ ∂I4, φ(x , t) ∈ T ⇐⇒ t = 1/2;

3 the sweepout t 7→ φ(1/2,1/2,1/2,1/2,1/2, t) is non-trivial.


L([φ]) > 4π = maxx∈∂I5

area(φ(x)).





L([φ]) > 4π = maxx∈∂I5

area(φ(x)).





L([φ]) > 4π = maxx∈∂I5

area(φ(x)).
Min-max Theory

CorollaryFor φ as in previous theorem, maxx∈I5 area(φ(x)) ≥ 2π2.

Idea:L([φ]) > max

x∈∂I5area(φ(x)) = 4π

(Pitts’ Thm) =⇒ ∃Σ minimal: L([φ]) = area(Σ) > 4π and index(Σ) ≤ 5

(Urbano’s Thm) =⇒ Σ = Clifford torus =⇒ maxx∈I5

area(φ(x)) ≥ L([φ]) = 2π2.
Min-max Theory


Idea:L([φ]) > max




area(φ(x)) ≥ L([φ]) = 2π2.
Min-max Theory


Idea:L([φ]) > max




area(φ(x)) ≥ L([φ]) = 2π2.
Min-max Theory


Idea:L([φ]) > max




area(φ(x)) ≥ L([φ]) = 2π2.
Sketch of proof of Theorem:

• Assume L([φ]) = 4π = maxx∈Ik area (ψ(x)) for some ψ ∈ [φ].

• Set K = ψ−1(T ), where ∂K ⊂ ∂I4 × {1/2}.

(A) [∂K ] = [∂I4 × {1/2}] in H3(∂I4 × {1/2},Z)

φ = ψ on ∂K and ψ : K → T ≈ S3

=⇒ degree(φ|∂I4×{1/2}) = degree(ψ|∂K ) = 0

(B) [∂K ] = 0 in H3(∂I4 × {1/2},Z)ψ ◦ c non-trivial sweepout with max

t∈Iarea (ψ ◦ c(t)) ≤ max

x∈Ikarea (ψ(x)) = 4π

=⇒ ψ ◦ c(I) ∩ T 6= ∅ =⇒ c(I) ∩ K 6= ∅.

• Assume L([φ]) = 4π = maxx∈Ik area (ψ(x)) for some ψ ∈ [φ].• Set K = ψ−1(T ), where ∂K ⊂ ∂I4 × {1/2}.

(A) [∂K ] = [∂I4 × {1/2}] in H3(∂I4 × {1/2},Z)






=⇒ ψ ◦ c(I) ∩ T 6= ∅ =⇒ c(I) ∩ K 6= ∅.


(A) [∂K ] = [∂I4 × {1/2}] in H3(∂I4 × {1/2},Z)






=⇒ ψ ◦ c(I) ∩ T 6= ∅ =⇒ c(I) ∩ K 6= ∅.


(A) [∂K ] = [∂I4 × {1/2}] in H3(∂I4 × {1/2},Z)






=⇒ ψ ◦ c(I) ∩ T 6= ∅ =⇒ c(I) ∩ K 6= ∅.


(A) [∂K ] = [∂I4 × {1/2}] in H3(∂I4 × {1/2},Z)



(B) [∂K ] = 0 in H3(∂I4 × {1/2},Z)

ψ ◦ c non-trivial sweepout with maxt∈I

area (ψ ◦ c(t)) ≤ maxx∈Ik

area (ψ(x)) = 4π

=⇒ ψ ◦ c(I) ∩ T 6= ∅ =⇒ c(I) ∩ K 6= ∅.


(A) [∂K ] = [∂I4 × {1/2}] in H3(∂I4 × {1/2},Z)






=⇒ ψ ◦ c(I) ∩ T 6= ∅ =⇒ c(I) ∩ K 6= ∅.
Theorem (Marques-N.)If Σ ⊂ S3 has positive genus thenW(Σ) ≥ 2π2.

1 Given v ∈ B4 consider Fv ∈ Conf (S3), x 7→ 1−|v |2

|x−v |2 (x − v)− v

2 C : B4 × [−π, π]→ Z2(S3), (v , t) 7→ surface at distance t from Fv (Σ)3 (Ros, 99) area(C(v , t)) ≤ W(C(v ,0)) =W(Fv (Σ)) =W(Σ)

4 Reparametrize C to obtain φ : B4 × [−π, π]→ Z2(S3) continuous with

• image(φ) = image(C);• φ(S3 × [−π, π]) ⊂ R and φ(x , t) ∈ T ⇐⇒ t = 0;• t 7→ φ(0, t), π ≤ t ≤ π, is non-trivial sweepout

5 degree(φ|S3×{0}) = genus of Σ!6 Corollary to our Min-max Thm can be applied and so

2π2 ≤ max(v ,t)∈B4×[−π,π]

area(φ(x)) = max(v ,t)∈B4×[−π,π]

area(C(v , t)) ≤ W(Σ)


|x−v |2 (x − v)− v





2π2 ≤ max(v ,t)∈B4×[−π,π]




|x−v |2 (x − v)− v

2 C : B4 × [−π, π]→ Z2(S3), (v , t) 7→ surface at distance t from Fv (Σ)

3 (Ros, 99) area(C(v , t)) ≤ W(C(v ,0)) =W(Fv (Σ)) =W(Σ)




2π2 ≤ max(v ,t)∈B4×[−π,π]




|x−v |2 (x − v)− v





2π2 ≤ max(v ,t)∈B4×[−π,π]




|x−v |2 (x − v)− v





2π2 ≤ max(v ,t)∈B4×[−π,π]




|x−v |2 (x − v)− v




5 degree(φ|S3×{0}) = genus of Σ!

6 Corollary to our Min-max Thm can be applied and so

2π2 ≤ max(v ,t)∈B4×[−π,π]




|x−v |2 (x − v)− v





2π2 ≤ max(v ,t)∈B4×[−π,π]


Theorem (Agol-Marques-N.)If (γ1, γ2) link in S3 has lk(γ1, γ2) = ±1, then E(γ1, γ2) ≥ 2π2.

• Given (σ1, σ2) ⊂ R4 consider G(σ1, σ2) : S1 × S1 → S3

G(γ1, γ2)(s, t) =(σ1(s)− σ2(t))|σ1(s)− σ2(t)|

.

• area(G(σ1, σ2)) ≤ E(γ1, γ2).• Given v ∈ B4, Fv (x) = x−v|x−v |2 ∈ Conf (R

4). Fv (B1(0)) = Br(v)(c(v)).

1 φ : B4 × [0,+∞]→ Z2(S3),

φ(v , λ) = G(Fv (γ1), λ(Fv (γ2)− c(v)) + c(v)).

2 area(φ(v , λ)) ≤ E(φ(v ,1)) = E(Fv (γ1),Fv (γ2)) = E(γ1, γ2).3 Given v ∈ S3,

φ(v ,1) = lk(γ1, γ2)∂Bπ/2(−v) =⇒ degree(φ|S3×{1}) = lk(γ1, γ2) 6= 0

4 Can apply our Min-max Thm to conclude

2π2 ≤ max(v ,t)∈B4×[0,+∞] area(φ(v , t)) ≤ E(γ1, γ2).


G(γ1, γ2)(s, t) =(σ1(s)− σ2(t))|σ1(s)− σ2(t)|

.


4). Fv (B1(0)) = Br(v)(c(v)).

1 φ : B4 × [0,+∞]→ Z2(S3),







G(γ1, γ2)(s, t) =(σ1(s)− σ2(t))|σ1(s)− σ2(t)|

.

• area(G(σ1, σ2)) ≤ E(γ1, γ2).

• Given v ∈ B4, Fv (x) = x−v|x−v |2 ∈ Conf (R4). Fv (B1(0)) = Br(v)(c(v)).

1 φ : B4 × [0,+∞]→ Z2(S3),







G(γ1, γ2)(s, t) =(σ1(s)− σ2(t))|σ1(s)− σ2(t)|

.


4). Fv (B1(0)) = Br(v)(c(v)).

1 φ : B4 × [0,+∞]→ Z2(S3),







G(γ1, γ2)(s, t) =(σ1(s)− σ2(t))|σ1(s)− σ2(t)|

.


4). Fv (B1(0)) = Br(v)(c(v)).

1 φ : B4 × [0,+∞]→ Z2(S3),







G(γ1, γ2)(s, t) =(σ1(s)− σ2(t))|σ1(s)− σ2(t)|

.


4). Fv (B1(0)) = Br(v)(c(v)).

1 φ : B4 × [0,+∞]→ Z2(S3),


2 area(φ(v , λ)) ≤ E(φ(v ,1)) = E(Fv (γ1),Fv (γ2)) = E(γ1, γ2).

3 Given v ∈ S3,





G(γ1, γ2)(s, t) =(σ1(s)− σ2(t))|σ1(s)− σ2(t)|

.


4). Fv (B1(0)) = Br(v)(c(v)).

1 φ : B4 × [0,+∞]→ Z2(S3),







G(γ1, γ2)(s, t) =(σ1(s)− σ2(t))|σ1(s)− σ2(t)|

.


4). Fv (B1(0)) = Br(v)(c(v)).

1 φ : B4 × [0,+∞]→ Z2(S3),

Min-max Theory, Willmore conjecture, and Energy of links - 0.3in...

Documents

Transcript of Min-max Theory, Willmore conjecture, and Energy of links - 0.3in...