how much does it costto evert the sphere0(imm(s2,r3)) = {1} i.e. two arbitrary c2 immersions of s2...

How Much Does It Cost...To Evert The Sphere ?Tristan Rivière

Tristan Rivière J.M.Coron 60th Birthday. I.H.P. 23/06/16 1

Fenchel’s Theorem

Theorem. [Fenchel 1929] Let � be a simple closed C2 curve in Rm,we have ⁄

�Ÿ dl� Ø 2fi

with equality iff � is planar and convex.


Fenchel’s TheoremTheorem. [Fenchel 1929] Let � be a simple closed C2 curve in Rm,we have ⁄

�Ÿ dl� Ø 2fi

with equality iff � is planar and convex.

R� k dl > 2⇡

R� k dl = 2⇡


Milnor’s Theorem

Theorem. [Milnor 1950] Let � be a simple closed C2 curve in R3,Assume � is knotted then we have⁄

�Ÿ dl > 4fi


Milnor’s TheoremTheorem. [Milnor 1950] Let � be a simple closed C2 curve in R3,Assume � is knotted then we have⁄

�Ÿ dl > 4fi

R� k dl > 4⇡


Euler’s Elastica

A curve “ in R2 is called an Euler Elastica if it is an equilibrium ofthe elastic energy


Euler’s ElasticaA curve “ in R2 is called an Euler Elastica if it is an equilibrium ofthe elastic energy

E(�) :=R�

2 dl

Figure : A model for Elastic Energy of Rods


Sophie Germain’s Generalization of Euler Elastica

She proposes the following free energy

W (S) :=⁄

S

-----1

2fi

⁄

normal planesŸ

-----

2

dvolS =⁄

SH2 dvolS

ddt

Area(S) = ≠ 2⁄

SH w dvolS and

ddt

Length(“) = ≠⁄

“Ÿ w dl


Willmore Theorem.Theorem [Willmore 1965] For any closed surface S µ R3

W (S) Ø 4fi

with equality iff S is a round sphere.

RS H

2 dvolS > 4⇡

RS H

2 dvolS = 4⇡


Li Yau Willmore Lower BoundTheorem [Li-Yau 1982] For any closed surface S

W (S) Ø 4fiN

where N is the maximal number of self-intersections.

RS H

2 dvolS � 8⇡

Corollary If W (S) < 8fi the surface is embedded.


The Willmore ConjectureTheorem [Marques-Neves 2011] For any S closed and genus(S) ”= 0

W (S) Ø 2fi2

with = iff S is conformally congruent to the Willmore Torus .

p2

1x

y

z


Everting The Sphere ?Theorem [Smale 1950] fi0(Imm(S2,R3)) = {1} i.e. two arbitrary C2

immersions of S2 into R3 are regular homotopic.

Figure : Sphere Eversion


How Much Does It Cost to Evert S2 in R3 ?

Let ⌃ be the set of elements in C0([0, 1], Imm(S2,R3)) realizing aneversion. Compute

Ev(S2) := infSt œ⌃

maxtœ[0,1]

W (St ) :=⁄

St

H2t dvolSt

Is it achieved by some critical point of W i.e. a so called WillmoreSphere?


What do we know about Willmore Spheres in R3 ?

Theorem [Bryant 1984] The Willmore immersions of S2 into R3 arethe inversions of the complete minimal surfaces in R3 of genus 0with flat ends.

For each such an immersion one has

W (S) = 4fiN ,

where N is the number of ends of the minimal surface. The first fourpossible N are

N = 1, 4, 6, 8

⇤





W (S) = 4fiN ,

where N is the number of ends of the minimal surface.

The first fourpossible N are

N = 1, 4, 6, 8

⇤





W (S) = 4fiN ,

where N is the number of ends of the minimal surface. The first fourpossible N are

N = 1, 4, 6, 8

⇤


Euler’s Elastica

A curve � in R3 is called an Euler Elastica if it is an equilibrium ofthe elastic energy ⁄

�Ÿ2 dl

An Euler Elastica satisfies in arc-length parametrization

2 Ÿ + Ÿ3 = 0

Multiplying by Ÿ and integrating gives

Ÿ2 +Ÿ4

4= C0 ≈∆ Ÿ

ÒC0 ≠ Ÿ4

4

= ±1

can be integrated using elliptic integrals.


Euler’s ElasticaA curve � in R3 is called an Euler Elastica if it is an equilibrium ofthe elastic energy ⁄

�Ÿ2 dl

An Euler Elastica satisfies in arc-length parametrization

2 Ÿ + Ÿ3 = 0

Multiplying by Ÿ and integrating gives

Ÿ2 +Ÿ4

4= C0 ≈∆ Ÿ

ÒC0 ≠ Ÿ4

4

= ±1

can be integrated using elliptic integrals.


The First Variation of Willmore Energy.

For normal perturbations w = w n we haveddt

dvolg = ≠ 2H w dvolg

andddt

H =12

Ë�gw + |dn|2g w

È

Theorem [Blaschke, Thomsen 1923] The first variation of W givesddt

W (St ) =⁄

⌃

Ë�gH + 2 H (H2 ≠ K )

Èw dvolg

In particular S is a critical point of W (i.e. Willmore) iff

�gH + 2 H (H2 ≠ K ) = 0

⇤compare with Euler Elastica 2 Ÿ + Ÿ3 = 0.


The First Variation of Willmore Energy.For normal perturbations w = w n we have

ddt


andddt

H =12

Ë�gw + |dn|2g w

È


W (St ) =⁄

⌃

Ë�gH + 2 H (H2 ≠ K )

Èw dvolg


�gH + 2 H (H2 ≠ K ) = 0




ddt


andddt

H =12

Ë�gw + |dn|2g w

È


W (St ) =⁄

⌃

Ë�gH + 2 H (H2 ≠ K )

Èw dvolg


�gH + 2 H (H2 ≠ K ) = 0

⇤

compare with Euler Elastica 2 Ÿ + Ÿ3 = 0.



ddt


andddt

H =12

Ë�gw + |dn|2g w

È


W (St ) =⁄

⌃

Ë�gH + 2 H (H2 ≠ K )

Èw dvolg


�gH + 2 H (H2 ≠ K ) = 0



Infinitesimal Symmetries

Let � : [0, 1] æ Rm and

F (�) =⁄ 1

0f

A

�,d�dx

,d2�

dx2

B

dx

Y : Rm æ Rm is an infinitesimal symmetry of f (q, p, a) if

’� ’ t > 0 f

A

Ït ¶ �,dÏt ¶ �

dx,

d2Ït ¶ �

dx2

B

© f

A

�,d�dx

,d2�

dx2

B

where Ït is the flow of Y . We then have

”F (�) +ddx

C

ˆpf Y ≠ d(ˆaf )dt

Y + ˆafdYdx

D

© 0


Infinitesimal SymmetriesLet � : [0, 1] æ Rm and

F (�) =⁄ 1

0f

A

�,d�dx

,d2�

dx2

B

dx


’� ’ t > 0 f

A

Ït ¶ �,dÏt ¶ �

dx,

d2Ït ¶ �

dx2

B

© f

A

�,d�dx

,d2�

dx2

B


”F (�) +ddx

C


Y + ˆafdYdx

D

© 0



F (�) =⁄ 1

0f

A

�,d�dx

,d2�

dx2

B

dx


’� ’ t > 0 f

A

Ït ¶ �,dÏt ¶ �

dx,

d2Ït ¶ �

dx2

B

© f

A

�,d�dx

,d2�

dx2

B

where Ït is the flow of Y .

We then have

”F (�) +ddx

C


Y + ˆafdYdx

D

© 0



F (�) =⁄ 1

0f

A

�,d�dx

,d2�

dx2

B

dx


’� ’ t > 0 f

A

Ït ¶ �,dÏt ¶ �

dx,

d2Ït ¶ �

dx2

B

© f

A

�,d�dx

,d2�

dx2

B


”F (�) +ddx

C


Y + ˆafdYdx

D

© 0


Noether’s Theorem

Theorem [Noether 1918] Assume f (q, p, a) has infinitesimalsymmetry Y and � be a critical point for F then

ddx

C


Y + ˆafdYdx

D

© 0 .

Application. Euler elastica. In normal parametrization

Translation invariance =∆ ddx

C

2d Ÿ

dx+ 3 |Ÿ|2 d�

dx

D

= 0 .

Rotation invariance =∆ ddx

C

� · C0 + 2d�dx

· Ÿ

D

= 0 .

where C0 := 2 ˙Ÿ + 3 Ÿ2 ˙� Y := A resp. Y := B · �


Noether’s TheoremTheorem [Noether 1918] Assume f (q, p, a) has infinitesimalsymmetry Y and � be a critical point for F then

ddx

C


Y + ˆafdYdx

D

© 0 .



C

2d Ÿ

dx+ 3 |Ÿ|2 d�

dx

D

= 0 .


C

� · C0 + 2d�dx

· Ÿ

D

= 0 .




ddx

C


Y + ˆafdYdx

D

© 0 .

Application. Euler elastica.

In normal parametrization


C

2d Ÿ

dx+ 3 |Ÿ|2 d�

dx

D

= 0 .


C

� · C0 + 2d�dx

· Ÿ

D

= 0 .




ddx

C


Y + ˆafdYdx

D

© 0 .



C

2d Ÿ

dx+ 3 |Ÿ|2 d�

dx

D

= 0 .


C

� · C0 + 2d�dx

· Ÿ

D

= 0 .




ddx

C


Y + ˆafdYdx

D

© 0 .



C

2d Ÿ

dx+ 3 |Ÿ|2 d�

dx

D

= 0 .


C

� · C0 + 2d�dx

· Ÿ

D

= 0 .

where C0 := 2 ˙Ÿ + 3 Ÿ2 ˙�

Y := A resp. Y := B · �



ddx

C


Y + ˆafdYdx

D

© 0 .



C

2d Ÿ

dx+ 3 |Ÿ|2 d�

dx

D

= 0 .


C

� · C0 + 2d�dx

· Ÿ

D

= 0 .

where C0 := 2 ˙Ÿ + 3 Ÿ2 ˙� Y := A resp. Y := B · �Tristan Rivière J.M.Coron 60th Birthday. I.H.P. 23/06/16 15

Noether Principle for Willmore - Translations

Let � be Willmore. Take local conformal coordinates.

Invariance by translations : take Y := A and obtain

divËÒH ≠ 2 H Òn ≠ H2 Ò�

È= 0

Ì

÷ L s. t. ÒH ≠ 2 H Òn ≠ H2 Ò� = Ò‹L

Compare with Euler Elastica

ddx

C

2d Ÿ

dx+ 3 |Ÿ|2 d�

dx

D

= 0 ≈∆ 2d Ÿ

dx+ 3 |Ÿ|2 d�

dx= C0


Noether Principle for Willmore - TranslationsLet � be Willmore. Take local conformal coordinates.

Invariance by translations : take Y := A and obtain

divËÒH ≠ 2 H Òn ≠ H2 Ò�

È= 0

Ì

÷ L s. t. ÒH ≠ 2 H Òn ≠ H2 Ò� = Ò‹L


ddx

C

2d Ÿ

dx+ 3 |Ÿ|2 d�

dx

D

= 0 ≈∆ 2d Ÿ

dx+ 3 |Ÿ|2 d�

dx= C0


Noether Principle for Willmore - Rotations

Let � be Willmore. Take local conformal coordinates.Invariance by rotations : take Y := B · � and obtain

divË� · Ò‹L + 2 Ò� · H

È= 0

whereÒ‹L := ÒH ≠ 2 H Òn ≠ H2 Ò�


ddx

C

� · C0 + 2d�dx

· Ÿ

D

= 0 .

where C0 := 2 ˙Ÿ + 3 Ÿ2 ˙�




Invariance by rotations : take Y := B · � and obtain

divË� · Ò‹L + 2 Ò� · H

È= 0



ddx

C

� · C0 + 2d�dx

· Ÿ

D

= 0 .

where C0 := 2 ˙Ÿ + 3 Ÿ2 ˙�



Let � be Willmore. Take local conformal coordinates.Invariance by rotations : take Y := B · � and obtain

divË� · Ò‹L + 2 Ò� · H

È= 0



ddx

C

� · C0 + 2d�dx

· Ÿ

D

= 0 .

where C0 := 2 ˙Ÿ + 3 Ÿ2 ˙�


Noether Principle for Willmore - Dilations


Invariance by dilations : take Y := � and obtain

divË� · Ò‹L

È= 0


Compare with Euler Elastica. No invariance by dilation.

ddx

Ë� · C0

È= Ÿ2 where C0 := 2 ˙Ÿ + 3 Ÿ2 ˙�

The conservation law is replaced by a monotonicity formula.


Willmore Equation as a second order System.

Introduce

ÒS := L · Ò� and ÒR := L ◊ Ò� + 2 H Ò�

then Y______]

______[

�S = Òn · Ò‹R

�R = Òn ◊ Ò‹R + Òn Ò‹S

�� = Ò� ◊ Ò‹R + Ò�Ò‹SThe last line is a conservation law related to the invariance underinversion ¶ translation :

Y := |�|2 A ≠ 2 � · A �

Bernard 2013


Willmore Equation as a second order System.Introduce

ÒS := L · Ò� and ÒR := L ◊ Ò� + 2 H Ò�

then Y______]

______[

�S = Òn · Ò‹R



Y := |�|2 A ≠ 2 � · A �

Bernard 2013



ÒS := L · Ò� and ÒR := L ◊ Ò� + 2 H Ò�

then Y______]

______[

�S = Òn · Ò‹R


�� = Ò� ◊ Ò‹R + Ò�Ò‹S

The last line is a conservation law related to the invariance underinversion ¶ translation :

Y := |�|2 A ≠ 2 � · A �

Bernard 2013



ÒS := L · Ò� and ÒR := L ◊ Ò� + 2 H Ò�

then Y______]

______[

�S = Òn · Ò‹R



Y := |�|2 A ≠ 2 � · A �

Bernard 2013Tristan Rivière J.M.Coron 60th Birthday. I.H.P. 23/06/16 19

Regularity of Weak Willmore Immersions

Theorem [R. 2008] Let � be a weak immersion critical point of Wthen, in conformal coordinates, it is real analytic.

Proof. Use the existence of conformal coordinates + Noether andrewrite Willmore PDE in the second order system

Y______]

______[

�S = Òn · Ò‹R


�� = Ò� ◊ Ò‹R + Ò�Ò‹S

Apply Wente Integrability by compensation theory to get H œ Lp forsome p > 2. Bootstrap.


Regularity of Weak Willmore ImmersionsTheorem [R. 2008] Let � be a weak immersion critical point of Wthen, in conformal coordinates, it is real analytic.


Y______]

______[

�S = Òn · Ò‹R


�� = Ò� ◊ Ò‹R + Ò�Ò‹S




Proof.

Use the existence of conformal coordinates + Noether andrewrite Willmore PDE in the second order system

Y______]

______[

�S = Òn · Ò‹R


�� = Ò� ◊ Ò‹R + Ò�Ò‹S





Y______]

______[

�S = Òn · Ò‹R


�� = Ò� ◊ Ò‹R + Ò�Ò‹S





Y______]

______[

�S = Òn · Ò‹R


�� = Ò� ◊ Ò‹R + Ò�Ò‹S

Apply Wente Integrability by compensation theory to get H œ Lp forsome p > 2.

Bootstrap.




Y______]

______[

�S = Òn · Ò‹R


�� = Ò� ◊ Ò‹R + Ò�Ò‹S



The Origins of Minmax Methods

Birkhoff 1917. sweepouts of N2 : u œ C0([0, 1], W 1,2(S1, N2)) s.t.

u([0, 1] ◊ S1) generates H2(N2,Z)


The Origins of Minmax MethodsBirkhoff 1917.

sweepouts of N2 : u œ C0([0, 1], W 1,2(S1, N2)) s.t.



The Origins of Minmax MethodsBirkhoff 1917. sweepouts of N2 :

u œ C0([0, 1], W 1,2(S1, N2)) s.t.



The Origins of Minmax MethodsBirkhoff 1917. sweepouts of N2 : u œ C0([0, 1], W 1,2(S1, N2)) s.t.



The Origins of Minmax MethodsBirkhoff 1917. sweepouts of N2 : u œ C0([0, 1], W 1,2(S1, N2)) s.t.


[0, 1]⇥ S1

u(s, ·)

N2


Birkhoff Min-Max Method

—0 := infu(s,·) sweepout

maxsœ[0,1]

⁄

S1

----dudt

(s, t)----2

dt

is achieved by a closed geodesic in normal parametrization.

Main tool : Curve shortening process :

Each sweepout is replaced by a more regular one made of portionsof geodesics. Transform the Min-Max into a more compact one.Constructive proof.




maxsœ[0,1]

⁄

S1

----dudt

(s, t)----2

dt



Each sweepout is replaced by a more regular one made of portionsof geodesics.

Transform the Min-Max into a more compact one.Constructive proof.




maxsœ[0,1]

⁄

S1

----dudt

(s, t)----2

dt



Each sweepout is replaced by a more regular one made of portionsof geodesics. Transform the Min-Max into a more compact one.

Constructive proof.




maxsœ[0,1]

⁄

S1

----dudt

(s, t)----2

dt



Each sweepout is replaced by a more regular one made of portionsof geodesics. Transform the Min-Max into a more compact one.Constructive proof.


Viscosity Approach to Min-Max - General Scheme

1) Consider Immersions � of S1 into N2.Choose a Relaxation ofthe length of the form

L‡(�) = Length(�) + ‡2⁄

S1curvature terms dlg�

satisfying the Palais-Smale Condition to which Mountain PathLemma can be applied.

⁄

S1curvature terms dlg� is called the “smoother”

2) Make ‡ ≠æ 0 · · ·



1) Consider Immersions � of S1 into N2.

Choose a Relaxation ofthe length of the form

L‡(�) = Length(�) + ‡2⁄



⁄


2) Make ‡ ≠æ 0 · · ·




L‡(�) = Length(�) + ‡2⁄



⁄


2) Make ‡ ≠æ 0 · · ·




L‡(�) = Length(�) + ‡2⁄


satisfying the Palais-Smale Condition

to which Mountain PathLemma can be applied.

⁄


2) Make ‡ ≠æ 0 · · ·




L‡(�) = Length(�) + ‡2⁄



⁄


2) Make ‡ ≠æ 0 · · ·




L‡(�) = Length(�) + ‡2⁄



⁄


2) Make ‡ ≠æ 0 · · ·Tristan Rivière J.M.Coron 60th Birthday. I.H.P. 23/06/16 23

Viscosity Approach - A first difficulty

Proposition There exists �‡ critical point of

L‡(�) := Length(�(S1)) + ‡2⁄

S1Ÿ2�

dl

in normal parametrization s.t.

d�‡

dtÔ “Œ weakly in (LŒ)ú

butd�‡

dtnowhere strongly converge in L1

and“Œ is not a geodesic !


Viscosity Approach - A first difficultyProposition There exists �‡ critical point of

L‡(�) := Length(�(S1)) + ‡2⁄

S1Ÿ2�

dl

in normal parametrization s.t.

d�‡

dtÔ “Œ weakly in (LŒ)ú

butd�‡

dtnowhere strongly converge in L1

and“Œ is not a geodesic !


Viscosity Approach - A second difficulty.

Consider sweepouts of N2 by C1 immersions of S1.

—‡ := inf�(s,·) sweepout

maxsœ[0,1]

Length(�(s, S1)) + ‡2⁄

S1Ÿ2�

dl

is achieved by �‡ (classical min-max theory).Question : Do we have

lim‡æ0

Length (�‡) = —0 ≈∆ ‡2⁄

S1Ÿ2�‡

dl = o(1) ?

Struwe’s Monotonicity Trick 1988 There exists ‡k æ 0 s.t.

‡2k

⁄

S1Ÿ2�‡k

dl = o3

1log ‡k

4


Viscosity Approach - A second difficulty.Consider sweepouts of N2 by C1 immersions of S1.


maxsœ[0,1]

Length(�(s, S1)) + ‡2⁄

S1Ÿ2�

dl


lim‡æ0

Length (�‡) = —0 ≈∆ ‡2⁄

S1Ÿ2�‡

dl = o(1) ?


‡2k

⁄

S1Ÿ2�‡k

dl = o3

1log ‡k

4




maxsœ[0,1]

Length(�(s, S1)) + ‡2⁄

S1Ÿ2�

dl

is achieved by �‡ (classical min-max theory).

Question : Do we have

lim‡æ0

Length (�‡) = —0 ≈∆ ‡2⁄

S1Ÿ2�‡

dl = o(1) ?


‡2k

⁄

S1Ÿ2�‡k

dl = o3

1log ‡k

4




maxsœ[0,1]

Length(�(s, S1)) + ‡2⁄

S1Ÿ2�

dl


lim‡æ0

Length (�‡) = —0

≈∆ ‡2⁄

S1Ÿ2�‡

dl = o(1) ?


‡2k

⁄

S1Ÿ2�‡k

dl = o3

1log ‡k

4




maxsœ[0,1]

Length(�(s, S1)) + ‡2⁄

S1Ÿ2�

dl


lim‡æ0

Length (�‡) = —0 ≈∆ ‡2⁄

S1Ÿ2�‡

dl = o(1) ?


‡2k

⁄

S1Ÿ2�‡k

dl = o3

1log ‡k

4


Viscosity Approach to Minmax Geodesics.

Theorem. Michelat, R. 2015 Let


maxsœ[0,1]

Length(�(s, S1))

then there exists �n critical points of L‡n realizing

—‡n := inf�(s,·) sweepout

maxsœ[0,1]

Length(�(s, S1)) + ‡2n

⁄

S1Ÿ2�

dl

such that

lim‡æ0

Length (�‡n ) = —0 , ˙�n æ ˙�Œ. a.e

and �Œ(S1) is a geodesic.


Viscosity Approach to Minmax Geodesics.Theorem. Michelat, R. 2015 Let


maxsœ[0,1]

Length(�(s, S1))

then there exists �n critical points of L‡n realizing

—‡n := inf�(s,·) sweepout

maxsœ[0,1]

Length(�(s, S1)) + ‡2n

⁄

S1Ÿ2�

dl

such that

lim‡æ0

Length (�‡n ) = —0 , ˙�n æ ˙�Œ. a.e

and �Œ(S1) is a geodesic.Tristan Rivière J.M.Coron 60th Birthday. I.H.P. 23/06/16 26

Finding “Smoothers” for Willmore.

A first approach

F‡(�) := W (�) + ‡2⁄

⌃|dn|q dvolg�

is Palais Smale [Langer 1985] but is mixing 4th order andq≠harmonic theory.

A second approach

F‡(�) := W (�) + ‡2⁄

⌃|H�|q dvolg�

Applying Noether gives a 2nd order system with � as a mainoperator. However it is not Palais Smale .



A first approach

F‡(�) := W (�) + ‡2⁄

⌃|dn|q dvolg�

is Palais Smale [Langer 1985]

but is mixing 4th order andq≠harmonic theory.

A second approach

F‡(�) := W (�) + ‡2⁄

⌃|H�|q dvolg�




A first approach

F‡(�) := W (�) + ‡2⁄

⌃|dn|q dvolg�


A second approach

F‡(�) := W (�) + ‡2⁄

⌃|H�|q dvolg�




A first approach

F‡(�) := W (�) + ‡2⁄

⌃|dn|q dvolg�


A second approach

F‡(�) := W (�) + ‡2⁄

⌃|H�|q dvolg�

Applying Noether gives a 2nd order system with � as a mainoperator.

However it is not Palais Smale .



A first approach

F‡(�) := W (�) + ‡2⁄

⌃|dn|q dvolg�


A second approach

F‡(�) := W (�) + ‡2⁄

⌃|H�|q dvolg�



Frame Energies.

Let g0 such that ÷ – : ⌃ æ R

Kg0 = cte ,⁄

⌃dvolg0 = 1 and g� = e2– g0

Consider the Moser-Onofri Energy of –

O(�) :=12

⁄

⌃|d–|2g0

dvolg0+ Kg0

⁄

⌃– dvolg0≠ 2≠1 Kg0 log Area(�(⌃))

It is independent of the choice of – satisfying g� = e2– g0. For⌃ = S2 there is a group acting : M(S2). For ⌃ = T 2

W (�) + O(�) = inf12

⁄

T 2|de|2g0

dvolg0

where e are the orthonormal tangent frames associated to �.[Mondino, R. 2013].


Frame Energies.Let g0 such that ÷ – : ⌃ æ R

Kg0 = cte ,⁄



O(�) :=12

⁄

⌃|d–|2g0

dvolg0+ Kg0

⁄



W (�) + O(�) = inf12

⁄

T 2|de|2g0

dvolg0




Kg0 = cte ,⁄



O(�) :=12

⁄

⌃|d–|2g0

dvolg0+ Kg0

⁄


It is independent of the choice of – satisfying g� = e2– g0.

For⌃ = S2 there is a group acting : M(S2). For ⌃ = T 2

W (�) + O(�) = inf12

⁄

T 2|de|2g0

dvolg0




Kg0 = cte ,⁄



O(�) :=12

⁄

⌃|d–|2g0

dvolg0+ Kg0

⁄


It is independent of the choice of – satisfying g� = e2– g0. For⌃ = S2 there is a group acting : M(S2).

For ⌃ = T 2

W (�) + O(�) = inf12

⁄

T 2|de|2g0

dvolg0




Kg0 = cte ,⁄



O(�) :=12

⁄

⌃|d–|2g0

dvolg0+ Kg0

⁄



W (�) + O(�) = inf12

⁄

T 2|de|2g0

dvolg0



Viscous Approximation of Willmore

Introduce

F ‡(�) := W (�) + ‡2⁄

⌃(1 + |Hg� |2)2 dvolg� + |log ‡|≠1 O(�)

Theorem The functional F ‡ is Palais-Smale for the W 2,4 topology.

Theorem Critical points of F ‡ under fixed area constraint satisfy an‘≠regularity independent of ‡


Willmore Minmax Procedures for Spheres.

An Admissible family of immersions is a set A of C0 mappings fromX (top. Space) into the space E2 of W 2,4 immersions invariant underHomeo(E2) isotopic to the identity.

Theorem [R. 2015] Let A be an admissible family such that

—0 := infAœA

max�œA(X )

W (�)

Then there exists finitely many Willmore spheres ›1 · · · ›n away fromfinitely many points such that

—0 =nÿ

i=1W (›i ) ≠ 4fi N

where N œ N. ⇤


Willmore Minmax Procedures for Spheres.An Admissible family of immersions is a set A of C0 mappings fromX (top. Space) into the space E2 of W 2,4 immersions invariant underHomeo(E2) isotopic to the identity.


—0 := infAœA

max�œA(X )

W (�)


—0 =nÿ

i=1W (›i ) ≠ 4fi N

where N œ N. ⇤


Willmore Minmax Procedures for Spheres.An Admissible family of immersions is a set A of C0 mappings fromX (top. Space) into the space E2 of W 2,4 immersions invariant underHomeo(E2) isotopic to the identity.


—0 := infAœA

max�œA(X )

W (�)


—0 =nÿ

i=1W (›i ) ≠ 4fi N

where N œ N. ⇤Tristan Rivière J.M.Coron 60th Birthday. I.H.P. 23/06/16 30

Sphere Eversions.

A fundamental sphere eversion is a path “ of R3 immersions fromS2 into itself with two opposite orientations whose class infi1(Imm(S2)/Diff(S2)) = Z generates the group.

The cost of the sphere eversion is the number

—0 := inffund. eversions

max W (�(t , ·))

Theorem [R. 2015] there exists finitely many Willmore spheres›1 · · · ›n away from finitely many points such that

—0 =nÿ

i=1W (›i ) ≠ 4fi N

where N œ N. ⇤


Sphere Eversions.A fundamental sphere eversion is a path “ of R3 immersions fromS2 into itself with two opposite orientations whose class infi1(Imm(S2)/Diff(S2)) = Z generates the group.



max W (�(t , ·))


—0 =nÿ

i=1W (›i ) ≠ 4fi N

where N œ N. ⇤


Sphere Eversions.A fundamental sphere eversion is a path “ of R3 immersions fromS2 into itself with two opposite orientations whose class infi1(Imm(S2)/Diff(S2)) = Z generates the group.



max W (�(t , ·))


—0 =nÿ

i=1W (›i ) ≠ 4fi N

where N œ N. ⇤Tristan Rivière J.M.Coron 60th Birthday. I.H.P. 23/06/16 31

The Conjecture —0 = 16fi?

Is —0 achieved by the inversion of a genus 0 minimal surface with 4planar ends ? Kusner et al.

Picture by Bohle, Heller, Peters and Thomas


What is known about the 16fi≠conjecture

Theorem[Banchoff, Max 1981] Every sphere eversion has aquadruple point.Hence using Li and Yau we obtain —0 Ø 16fi

If our bubble tree would have no singularity we would get —0 œ 4fiN.



Theorem[Banchoff, Max 1981] Every sphere eversion has aquadruple point.

Hence using Li and Yau we obtain —0 Ø 16fi




Theorem[Banchoff, Max 1981] Every sphere eversion has aquadruple point.Hence using Li and Yau we obtain —0 Ø 16fi



Willmore as a “quasi-Morse Function” of Im(⌃,R3)

Question 1. IsW⌃ := inf

�œIm(⌃,R3)W (�)

achieved ?

Question 2. Let k > 0 and � œ fik (Im(S2,R3)). Is

—� = inf�(t ,·)ƒ�

maxtœSk

W (�(t , ·))

achieved by a Willmore Sphere ?

Theorem [Smale 1959] Im(S2,R3) ƒhom SO(3) ◊ ⌦2(SO(3)) and

fik (Im(S2,R3)) = fik (SO(3)) ◊ fik+2(SO(3))


Willmore as a “quasi-Morse Function” of Im(⌃,R3)Question 1. Is

W⌃ := inf�œIm(⌃,R3)

W (�)

achieved ?

Question 2. Let k > 0 and � œ fik (Im(S2,R3)). Is

—� = inf�(t ,·)ƒ�

maxtœSk

W (�(t , ·))

achieved by a Willmore Sphere ?

Theorem [Smale 1959] Im(S2,R3) ƒhom SO(3) ◊ ⌦2(SO(3)) and

fik (Im(S2,R3)) = fik (SO(3)) ◊ fik+2(SO(3))


how much does it costto evert the sphere0(imm(s2,r3)) = {1} i.e. two arbitrary c2 immersions of s2...

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