analysis aspects of willmore surfaces

103
3 juillet 2007, Paris. Des EDP au calcul scientifique. En l’honneur de Luc Tartar. ANALYSIS ASPECTS OF WILLMORE SURFACES. Tristan Rivi ` ere Departement Mathematik ETH Z ¨ urich (e–mail: [email protected]) (Homepage: http://www.math.ethz.ch/ riviere)

Upload: others

Post on 16-Jan-2022

1 views

Category:

Documents


0 download

TRANSCRIPT

3 juillet 2007, Paris. Des EDP au calcul scientifique. En l’honneur de Luc Tartar.

ANALYSIS ASPECTS OF WILLMORE SURFACES.

Tristan Riviere

Departement MathematikETH Zurich

(e–mail: [email protected])

(Homepage: http://www.math.ethz.ch/

riviere)

Curvatures for surfaces in� � . 1

Curvatures for surfaces in � � .

- � oriented closed surface in � � .

- � induced metric on � .

-� the unit normal to � (Gauss map).

Curvatures for surfaces in� � . 1

Curvatures for surfaces in � � .

- � oriented closed surface in � � .

- � induced metric on � .

-� the unit normal to � (Gauss map).The 2nd Fundamental form :

� � � � � � � ��

� � � � � � � � �� � � � � � � �� (1)

is bilinear symmetric from � � � � � into � � � the normal direction to � � � .

Curvatures for surfaces in� � . 1

Curvatures for surfaces in � � .

- � oriented closed surface in � � .

- � induced metric on � .

-� the unit normal to � (Gauss map).The 2nd Fundamental form :

� � � � � � � ��

� � � � � � � � �� � � � � � � �� (2)

is bilinear symmetric from � � � � � into � � � the normal direction to � � � .�

� � � � ��

� � �

� � ��

�� (3)

Curvatures for surfaces in� � . 1

Curvatures for surfaces in � � .

- � oriented closed surface in � � .

- � induced metric on � .

-� the unit normal to � (Gauss map).The 2nd Fundamental form :

� � � � � � � ��

� � � � � � � � �� � � � � � � �� (4)

is bilinear symmetric from � � � � � into � � � the normal direction to � � � .�

� � � � ��

� � �

� � ��

�� (5)

- Principal curvatures : � � and � �

- Mean curvature : � � �� � � � � � � .

- Mean curvature vector :

�� � �� .

- Gauss curvature : � � � � � � .

The Willmore energy of a closed surface in � � . 1

The Willmore energy of a closed surface in � � .

Willmore energy :

� � � �

�� � � � � � � � �

� �� � � � � � � � � � � � � (6)

The Willmore energy of a closed surface in � � . 1

The Willmore energy of a closed surface in � � .

Willmore energy :

� � � �

�� � � � � � � � �

� �� � � � � � � � � � � � � (7)

Gauss-Bonnet Theorem :

�� � � � � � � � � � � � � � � � � � � � � � (8)

The Willmore energy of a closed surface in � � . 1

The Willmore energy of a closed surface in � � .

Willmore energy :

� � � �

�� � � � � � � � �

� �� � � � � � � � � � � � � (9)

Gauss-Bonnet Theorem :

�� � � � � � � � � � � � � � � � � � � � � � (10)

Umbillic or anisotropy energy :

�� � � �

�� � � � � � � � � � �

� �� � � � � � � � � � � � � (11)

The Willmore energy of a closed surface in � � . 1

The Willmore energy of a closed surface in � � .

Willmore energy :

� � � �

�� � � � � � � � �

� �� � � � � � � � � � � � � (12)

Gauss-Bonnet Theorem :

�� � � � � � � � � � � � � � � � � � � � � � (13)

Umbillic or anisotropy energy :

�� � � �

�� � � � � � � � � � �

� �� � � � � � � � � � � � � (14)

Gauss equation :

� � � �� � � � �

��

� ��

� � � �� �� � � (15)

The Willmore energy of a closed surface in � � . 1

The Willmore energy of a closed surface in � � .

Willmore energy :

� � � �

�� � � � � � � � �

� �� � � � � � � � � � � � � (16)

Gauss-Bonnet Theorem :

�� � � � � � � � � � � � � � � � � � � � � � (17)

Umbillic or anisotropy energy :

�� � � �

�� � � � � � � � � � �

� �� � � � � � � � � � � � � (18)

Gauss equation :

� � � �� � � � �

��

� ��

� � � �� �� � � (19)

Conclusion : Modulo a topological invariant � � � is comparable to the homge-neous � ��

� norm of the Gauss map.

Willmore energy in various fields of science and technology. 1

Willmore energy in various fields of science and technology.

Willmore energy in various fields of science and technology. 1

Willmore energy in various fields of science and technology.� Conformal geometry (presumably the origin). Thomsen, Schadow 1923 -

Blaschke 1929. -...-Willmore 1965.

Willmore energy in various fields of science and technology. 1

Willmore energy in various fields of science and technology.� Conformal geometry (presumably the origin). Thomsen, Schadow 1923 -

Blaschke 1929. -...-Willmore 1965.

� General Relativity. Hawking 1968 mass of 2 spheres :

� � � � � � � � � �� �

� � � � � � �� � � �

�� � �� � � � � �

Willmore energy in various fields of science and technology. 1

Willmore energy in various fields of science and technology.� Conformal geometry (presumably the origin). Thomsen, Schadow 1923 -

Blaschke 1929. -...-Willmore 1965.

� General Relativity. Hawking 1968 mass of 2 spheres :

� � � � � � � � � �� �

� � � � � � �� � � �

�� � �� � � � � �

� Cell Biology. Helfrich 1973. Spontaneous curvature model for biomembranes,vesicles and smectic A-liquid crystals.

Willmore energy in various fields of science and technology. 1

Willmore energy in various fields of science and technology.� Conformal geometry (presumably the origin). Thomsen, Schadow 1923 -

Blaschke 1929. -...-Willmore 1965.

� General Relativity. Hawking 1968 mass of 2 spheres :

� � � � � � � � � �� �

� � � � � � �� � � �

�� � �� � � � � �

� Cell Biology. Helfrich 1973. Spontaneous curvature model for biomembranes,vesicles and smectic A-liquid crystals.

� Mechanic-Elasticity. Non-linear plate theory. � -limit elastic energy � � � � � �

(Friesecke, James, Muller 2001).

Willmore energy in various fields of science and technology. 1

Willmore energy in various fields of science and technology.� Conformal geometry (presumably the origin). Thomsen, Schadow 1923 -

Blaschke 1929. -...-Willmore 1965.

� General Relativity. Hawking 1968 mass of 2 spheres :

� � � � � � � � � �� �

� � � � � � �� � � �

�� � �� � � � � �

� Cell Biology. Helfrich 1973. Spontaneous curvature model for biomembranes,vesicles and smectic A-liquid crystals.

� Mechanic-Elasticity. Non-linear plate theory. � -limit elastic energy � � � � � �

(Friesecke, James, Muller 2001).

� Optical design. Rubinstein 1990.

� � �

Conformal invariance of Willmore energy. 1

Conformal invariance of Willmore energy.

Theorem. [Blaschke 1929] Let � be a closed oriented surface of � � , let � be aconformal diffeomorphism of � � � � � � then the following holds

� � � � � � � � � �

Conformal invariance of Willmore energy. 1

Conformal invariance of Willmore energy.

Theorem. [Blaschke 1929] Let � be a closed oriented surface of � � , let � be aconformal diffeomorphism of � � � � � � then the following holds

� � � � � � � � � �

� � � � � � where � � � � � � � �� � �� (inversion)

Generalization : Willmore energy of immersed surfaces in� � . 1

Generalization : Willmore energy of immersed surfaces in � � .

Generalization : Willmore energy of immersed surfaces in� � . 1

Generalization : Willmore energy of immersed surfaces in � � .

- � abstract oriented closed 2 dimensional manifold.-

�� smooth immersion of � in � � .

- � induced metric on � .-� � � oriented � � � space normal to the tangent 2-space at � (Gauss Map).- � �� : orthonormal projection onto the normal space given by� .

Generalization : Willmore energy of immersed surfaces in� � . 1

Generalization : Willmore energy of immersed surfaces in � � .

- � abstract oriented closed 2 dimensional manifold.-

�� smooth immersion of � in � � .

- � induced metric on � .-� � � oriented � � � space normal to the tangent 2-space at � (Gauss Map).- � �� : orthonormal projection onto the normal space given by� .

The 2nd Fundamental form :

� � � � � � � ��

� � � � � � � � �� � � ��

� � � � � � (20)

is bilinear symmetric.

Generalization : Willmore energy of immersed surfaces in� � . 1

Generalization : Willmore energy of immersed surfaces in � � .

- � abstract oriented closed 2 dimensional manifold.-

�� smooth immersion of � in � � .

- � induced metric on � .-� � � oriented � � � space normal to the tangent 2-space at � (Gauss Map).- � �� : orthonormal projection onto the normal space given by� .

The 2nd Fundamental form :

� � � � � � � ��

� � � � � � � � �� � � ��

� � � � � � (21)

is bilinear symmetric.

- Mean curvature vector :

�� � �� � � �

�� .

- Willmore energy : � ��

� �� � �

�� ��

� � � � � � .

Generalization : Willmore energy of immersed surfaces in� � . 1

Generalization : Willmore energy of immersed surfaces in � � .

- � abstract oriented closed 2 dimensional manifold.-

�� smooth immersion of � in � � .

- � induced metric on � .-� � � oriented � � � space normal to the tangent 2-space at � (Gauss Map).- � �� : orthonormal projection onto the normal space given by� .

The 2nd Fundamental form :

� � � � � � � ��

� � � � � � � � �� � � ��

� � � � � � (22)

is bilinear symmetric.

- Mean curvature vector :

�� � �� � � �

�� .

- Willmore energy : � ��

� �� � �

�� ��

� � � � � � .

In this talk we will restrict to � � � though most of the results presented are validin arbitrary dimension.

Willmore Immersions. 1

Willmore Immersions.

Willmore Immersions. 1

Willmore Immersions.

Definition : An immersion

�� : � � � � � is Willmore if ��

� � � �� � � � � � the

following holds

�� �� ��

� � ��

� � � � � � � �

Willmore Immersions. 1

Willmore Immersions.

Definition : An immersion

�� : � � � � � is Willmore if ��

� � � �� � � � � � the

following holds

�� �� ��

� � ��

� � � � � � � �

� Examples :-

�� is a minimal immersion :

�� � � .

Willmore Immersions. 1

Willmore Immersions.

Definition : An immersion

�� : � � � � � is Willmore if ��

� � � �� � � � � � the

following holds

�� �� ��

� � ��

� � � � � � � �

� Examples :-

�� is a minimal immersion :

�� � � .

-

�� is a composition of a minimal immersion and a conformal transformation.

Willmore Immersions. 1

Willmore Immersions.

Definition : An immersion

�� : � � � � � is Willmore if ��

� � � �� � � � � � the

following holds

�� �� ��

� � ��

� � � � � � � �

� Examples :-

�� is a minimal immersion :

�� � � .

-

�� is a composition of a minimal immersion and a conformal transformation.

- The round sphere � � . Consequence of the following inequality� � � �

�� � �� � � � � � � � �

holds for any closed surface � with equality iff � is a round sphere.

Willmore Immersions. 1

Willmore Immersions.

Definition : An immersion

�� : � � � � � is Willmore if ��

� � � �� � � � � � the

following holds

�� �� ��

� � ��

� � � � � � � �

� Examples :-

�� is a minimal immersion :

�� � � .

-

�� is a composition of a minimal immersion and a conformal transformation.

- The round sphere � � . Consequence of the following inequality� � � �

�� � �� � � � � � � � �

holds for any closed surface � with equality iff � is a round sphere.- The Willmore Torus � � .

� � � � � � ��

Willmore Immersions. 1

Willmore Immersions.

Definition : An immersion

�� : � � � � � is Willmore if ��

� � � �� � � � � � the

following holds

�� �� ��

� � ��

� � � � � � � �

� Examples :-

�� is a minimal immersion :

�� � � .

-

�� is a composition of a minimal immersion and a conformal transformation.

- The round sphere � � . Consequence of the following inequality� � � �

�� � �� � � � � � � � �

holds for any closed surface � with equality iff � is a round sphere.- The Willmore Torus � � .

� � � � � � ��

Minimizes W among all immersions of � � ? (Willmore conjecture).

Geometric and Analysis questions related to Willmore immersions. 1

Geometric and Analysis questions related to Willmoreimmersions.

Geometric and Analysis questions related to Willmore immersions. 1

Geometric and Analysis questions related to Willmoreimmersions.

� Explicit and exhaustive description of the space of Willmore immersions of agiven surface � .

Geometric and Analysis questions related to Willmore immersions. 1

Geometric and Analysis questions related to Willmoreimmersions.

� Explicit and exhaustive description of the space of Willmore immersions of agiven surface � .

� Is the set of Willmore surfaces below a certain level of energy strongly, weaklycompact ? (modulo the action of conformal transformations).

Geometric and Analysis questions related to Willmore immersions. 1

Geometric and Analysis questions related to Willmoreimmersions.

� Explicit and exhaustive description of the space of Willmore immersions of agiven surface � .

� Is the set of Willmore surfaces below a certain level of energy strongly, weaklycompact ? (modulo the action of conformal transformations).

� Does there exist minimizers of � among all immersions of a given surface �

?...identify these minimizers.

Geometric and Analysis questions related to Willmore immersions. 1

Geometric and Analysis questions related to Willmoreimmersions.

� Explicit and exhaustive description of the space of Willmore immersions of agiven surface � .

� Is the set of Willmore surfaces below a certain level of energy strongly, weaklycompact ? (modulo the action of conformal transformations).

� Does there exist minimizers of � among all immersions of a given surface �

?...identify these minimizers.

� Is there a notion of Weak Willmore immersions ? If so, what are the possiblesingularities ?

The Euler-Lagrange Equation of Thomsen and Schadow. 1

The Euler-Lagrange Equation of Thomsen and Schadow.

The Euler-Lagrange Equation of Thomsen and Schadow. 1

The Euler-Lagrange Equation of Thomsen and Schadow.

Theorem. [Thomsen, Schadow 1923]Let

�� be a smooth immersion of a surface � into � � .

�� is Willmore if and only if

the following Euler-Lagrange equation is satisfied

� � � � � � � � � �� � � � � � (23)

where � � is the negative Laplace Beltrami operator of the induced metric by

��

on � . �

The Euler-Lagrange Equation of Thomsen and Schadow. 1

The Euler-Lagrange Equation of Thomsen and Schadow.

Theorem. [Thomsen, Schadow 1923]Let

�� be a smooth immersion of a surface � into � � .

�� is Willmore if and only if

the following Euler-Lagrange equation is satisfied

� � � � � � � � � �� � � � � � (24)

where � � is the negative Laplace Beltrami operator of the induced metric by

��

on � . �

Functional analysis paradox : The formulation (24) of the Euler-Lagrange Equa-tion requires at least � to be in �

�� � �

which is more restrictive than the condition

� being in �� given by the finiteness of the Lagrangian � ��

� !!!

The Euler-Lagrange Equation in divergence form. 1

The Euler-Lagrange Equation in divergence form.

The Euler-Lagrange Equation in divergence form. 1

The Euler-Lagrange Equation in divergence form.

Theorem 1. [R. 2006] Let � be a smooth surface in � � , the following equation issatisfied

� � � � � � � � � �� � � � � � (25)

if and only if, in conformal coordinates, the following holds� � �

�� ��

� � � � �� ��

� � �� �

� � (26)

where the operators � � � , � and � � are taken with respect to the flat metric inthe conformal coordinates � � � � : � � � � � � � � � � � �

� � , � � � � � � � � and

� � � � � � � � � � . �

Divergence form for Schrodinger systems with antisymmetric potentials. 1

Divergence form for Schr odinger systems with antisymmetricpotentials.

Divergence form for Schrodinger systems with antisymmetric potentials. 1

Divergence form for Schr odinger systems with antisymmetricpotentials.

Theorem 2. [R. 2006] There exists a continuous operator

� �� � � � � � � � � � ��

� � ��

� � � � � � � � � � ��

� � � � � � �

� � � � � � � � �

(27)such that � � �

� � � � � � � is a solution of the Schrodinger system

� � � � � � � � in � � (28)

if and only if it satisfies

� � ��� �

� � � � � �

� � ��

� � � (29)

Conformally invariant Lagrangians and Schrodinger systems with antisymmetric potentials. 1

Conformally invariant Lagrangians and Schr odinger systemswith antisymmetric potentials.

Conformally invariant Lagrangians and Schrodinger systems with antisymmetric potentials. 1

Conformally invariant Lagrangians and Schr odinger systemswith antisymmetric potentials.

Theorem 3. [R. 2006] Let � be a Lagrangian on � ��

� � � � � � � of the form

� � � �

� � � � � � � � � � � (30)

with �� �

� � ��� � � � � � � � � � �� for some constant � . Assume moreover that � is

� � in � and � � in � . If � is conformal invariant :

� � � � � � � � �

for any conformal transformation � in the plane, then the Euler-Lagrange equationcan be written as a vectorial Schrodinger equation with antisymmetric potential.

Conformally invariant Lagrangians and Schrodinger systems with antisymmetric potentials. 1

Conformally invariant Lagrangians and Schr odinger systemswith antisymmetric potentials.

Theorem 3. [R. 2006] Let � be a Lagrangian on � ��

� � � � � � � of the form

� � � �

� � � � � � � � � � � (31)

with �� �

� � ��� � � � � � � � � � �� for some constant � . Assume moreover that � is

� � in � and � � in � . If � is conformal invariant :

� � � � � � � � �

for any conformal transformation � in the plane, then the Euler-Lagrange equationcan be written as a vectorial Schrodinger equation with antisymmetric potential.

Examples- Harmonic maps equations into riemannian manifolds.

Conformally invariant Lagrangians and Schrodinger systems with antisymmetric potentials. 1

Conformally invariant Lagrangians and Schr odinger systemswith antisymmetric potentials.

Theorem 3. [R. 2006] Let � be a Lagrangian on � ��

� � � � � � � of the form

� � � �

� � � � � � � � � � � (32)

with �� �

� � ��� � � � � � � � � � �� for some constant � . Assume moreover that � is

� � in � and � � in � . If � is conformal invariant :

� � � � � � � � �

for any conformal transformation � in the plane, then the Euler-Lagrange equationcan be written as a vectorial Schrodinger equation with antisymmetric potential.

Examples- Harmonic maps equations into riemannian manifolds- Prescribed mean curvature equations in riemannian manifolds

The Link between Willmore surfaces and Schrodinger equations with antisymmetric potentials. 1

The Link between Willmore surfaces and Schr odingerequations with antisymmetric potentials.

The Link between Willmore surfaces and Schrodinger equations with antisymmetric potentials. 1

The Link between Willmore surfaces and Schr odingerequations with antisymmetric potentials.

Fact 1: This last theorem extends to harmonic map equation intoLorentzian Manifolds which can hence be written in divergence form.

The Link between Willmore surfaces and Schrodinger equations with antisymmetric potentials. 1

The Link between Willmore surfaces and Schr odingerequations with antisymmetric potentials.

Fact 1: This last theorem extends to harmonic map equation intoLorentzian Manifolds which can hence be written in divergence form.

Fact 2 : [Blaschke 1929] A surface � is Willmore if and only if its conformal Gaussmap is harmonic into the space of oriented 2-spheres of � � isometric to � � � �

the Minkowski sphere of � ��

� .

The Link between Willmore surfaces and Schrodinger equations with antisymmetric potentials. 1

The Link between Willmore surfaces and Schr odingerequations with antisymmetric potentials.

Fact 1: This last theorem extends to harmonic map equation intoLorentzian Manifolds which can hence be written in divergence form.

Fact 2 : [Blaschke 1929] A surface � is Willmore if and only if its conformal Gaussmap is harmonic into the space of oriented 2-spheres of � � isometric to � � � �

the Minkowski sphere of � ��

� .

Conclusion : Willmore equation can then be written in divergence form !

Good things ...and insufficiencies of the divergence form of Willmore equation. 1

Good things ...and insufficiencies of the divergence form ofWillmore equation.

Good things ...and insufficiencies of the divergence form of Willmore equation. 1

Good things ...and insufficiencies of the divergence form ofWillmore equation.

� The Willmore Operator. For� � � ��

� � � � � � � and� � � �� � � � � � � denote

� ���� � � � �

�� ��� � � � � � � � � � �� � �� �

� (33)

Good things ...and insufficiencies of the divergence form of Willmore equation. 1

Good things ...and insufficiencies of the divergence form ofWillmore equation.

� The Willmore Operator. For� � � ��

� � � � � � � and� � � �� � � � � � � denote

� ���� � � � �

�� ��� � � � � � � � � � �� � �� �

� (34)

then � �� is formally self-adjoint. (Observe that

�� is Willmore � � ���� � � ).

Good things ...and insufficiencies of the divergence form of Willmore equation. 1

Good things ...and insufficiencies of the divergence form ofWillmore equation.

� The Willmore Operator. For� � � ��

� � � � � � � and� � � �� � � � � � � denote

� ���� � � � �

�� ��� � � � � � � � � � �� � �� �

� (35)

then � �� is formally self-adjoint. (Observe that

�� is Willmore � � ���� � � ).

� For any � � � � �� is continuous from � � into �� �

� � .

Good things ...and insufficiencies of the divergence form of Willmore equation. 1

Good things ...and insufficiencies of the divergence form ofWillmore equation.

� The Willmore Operator. For� � � ��

� � � � � � � and� � � �� � � � � � � denote

� ���� � � � �

�� ��� � � � � � � � � � �� � �� �

� (36)

then � �� is formally self-adjoint. (Observe that

�� is Willmore � � ���� � � ).

� For any � � � � �� is continuous from � � into �� �

� � .

� A-priori estimates : � � � � � s.t. if � � � � �� �� � � � then

��

� � ���

� � � � � � � ��

� � � � � � � � ���

� � � � � � � � (37)

Good things ...and insufficiencies of the divergence form of Willmore equation. 1

Good things ...and insufficiencies of the divergence form ofWillmore equation.

� The Willmore Operator. For� � � ��

� � � � � � � and� � � �� � � � � � � denote

� ���� � � � �

�� ��� � � � � � � � � � �� � �� �

� (38)

then � �� is formally self-adjoint. (Observe that

�� is Willmore � � ���� � � ).

� For any � � � � �� is continuous from � � into �� �

� � .

� A-priori estimates : � � � � � s.t. if � � � � �� �� � � � then

��

� � ���

� � � � � � � ��

� � � � � � � � ���

� � � � � � � � (39)

� This last 2 facts do not work anymore for � � � !

Willmore surfaces as the critical case for Willmore operator : the compactness issue 1

Willmore surfaces as the critical case for Willmore operato r :the compactness issue

Willmore surfaces as the critical case for Willmore operator : the compactness issue 1

Willmore surfaces as the critical case for Willmore operato r :the compactness issue

Question 1 : Modulo extraction of a subsequence is the weak limit of Willmoredisks still Willmore ?

Willmore surfaces as the critical case for Willmore operator : the compactness issue 1

Willmore surfaces as the critical case for Willmore operato r :the compactness issue

Question 1 : Modulo extraction of a subsequence is the weak limit of Willmoredisks still Willmore ?

Assume� � � � in � �

� and

�� � ��

� in ��

Willmore surfaces as the critical case for Willmore operator : the compactness issue 1

Willmore surfaces as the critical case for Willmore operato r :the compactness issue

Question 1 : Modulo extraction of a subsequence is the weak limit of Willmoredisks still Willmore ?

Assume� � � � in � �

� and

�� � ��

� in ��

Then

� � � � � ��� �

� � � � �� � ��� � � � � � � � �

�� �

� � � � �

� � � � � ��� ? ?

(40)

( � : space of Radon measures.)

Willmore surfaces as the critical case for Willmore operator : the regularity issue 1

Willmore surfaces as the critical case for Willmore operato r :the regularity issue

Willmore surfaces as the critical case for Willmore operator : the regularity issue 1

Willmore surfaces as the critical case for Willmore operato r :the regularity issue

Question 2 : Regularity of Willmore � � �

� surfaces ?

Willmore surfaces as the critical case for Willmore operator : the regularity issue 1

Willmore surfaces as the critical case for Willmore operato r :the regularity issue

Question 2 : Regularity of Willmore � � �

� surfaces ?

Definition : A � � �

� closed surface in � � is a compact subset of � � which realizesa � � �

� graph about every point. �

Willmore surfaces as the critical case for Willmore operator : the regularity issue 1

Willmore surfaces as the critical case for Willmore operato r :the regularity issue

Question 2 : Regularity of Willmore � � �

� surfaces ?

Definition : A � � �

� closed surface in � � is a compact subset of � � which realizesa � � �

� graph about every point. �

� �

�� �� �� � � � � � � � � �

Willmore surfaces as the critical case for Willmore operator : the regularity issue 1

Willmore surfaces as the critical case for Willmore operato r :the regularity issue

Question 2 : Regularity of Willmore � � �

� surfaces ?

Definition : A � � �

� closed surface in � � is a compact subset of � � which realizesa � � �

� graph about every point. �

� �

�� �� �� � � � � � � � � �

Theorem [Toro 1994, Muller-Sverak 1995] A � � �

� � graph in � � admits locally,about every point, a bilipschitz- � � �

� conformal parametrization

�� . �

Willmore surfaces as the critical case for Willmore operator : the regularity issue 1

Willmore surfaces as the critical case for Willmore operato r :the regularity issue

Question 2 : Regularity of Willmore � � �

� surfaces ?

Definition : A � � �

� closed surface in � � is a compact subset of � � which realizesa � � �

� graph about every point. �

� �

�� �� �� � � � � � � � � �

Theorem [Toro 1994, Muller-Sverak 1995] A � � �

� � graph in � � admits locally,about every point, a bilipschitz- � � �

� conformal parametrization

�� . �

From now on we can then assume :

�� � � ��

� � � � �

� ,� � � ��

� and theconformality condition� � � �

�� � � ��

Further conservation laws for Willmore surfaces. 1

Further conservation laws for Willmore surfaces.

Further conservation laws for Willmore surfaces. 1

Further conservation laws for Willmore surfaces.Theorem 4. [R. 2006] Let

�� be a conformal bilipschitz local parametrization of a

Willmore surface. Introduce

�� , satisfying

���

� � � ��

� � � � �� ��

� � �� � (41)

then the following holds

��

� � � ��

� � � and ��

� � ���

� � � �� � � ��

� � � � (42)

Further conservation laws for Willmore surfaces. 1

Further conservation laws for Willmore surfaces.Theorem 4. [R. 2006] Let

�� be a conformal bilipschitz local parametrization of a

Willmore surface. Introduce

�� , satisfying

���

� � � ��

� � � � �� ��

� � �� � (43)

then the following holds

��

� � � ��

� � � and ��

� � ���

� � � �� � � ��

� � � � (44)

Let

��

�� � ��

� � ��

��

� � ��

� ��

� � � � ��

� �

(45)

Further conservation laws for Willmore surfaces. 1

Further conservation laws for Willmore surfaces.Theorem 4. [R. 2006] Let

�� be a conformal bilipschitz local parametrization of a

Willmore surface. Introduce

�� , satisfying

���

� � � ��

� � � � �� ��

� � �� � (46)

then the following holds

��

� � � ��

� � � and ��

� � ���

� � � �� � � ��

� � � � (47)

Let

��

�� � ��

� � ��

��

� � ��

� ��

� � � � ��

� �

(48)

Then

��

� � ��

� � �� � � � � �� � � (49)

Recall notation : � � � �� � � � � � � � � � � � � � � .

Wente estimates and integrability by compensation. 1

Wente estimates and integrability by compensation.

Wente estimates and integrability by compensation. 1

Wente estimates and integrability by compensation.

Theorem [Wente 1969]

Let � and � be two functions in � ��

� � � � � � . Let � be the unique solution of

��� �

�� � �

� � � � � �� � � � � � � � � � � � � � � in � �

� � on � � � �

(50)

Then the following estimates hold true

� � � � � � � � � � � � � � � � � � � � � � � � � � � (51)

Wente estimates and integrability by compensation. 1

Wente estimates and integrability by compensation.

Theorem [Wente 1969, Tartar 1983]

Let � and � be two functions in � ��

� � � � � � . Let � be the unique solution of

��� �

�� � �

� � � � � �� � � � � � � � � � � � � � � in � �

� � on � � � �

(52)

Then the following estimates hold true

� � � � � � � � � � � � �� � � � � � � � � � � � � � � � (53)

Wente estimates and integrability by compensation. 1

Wente estimates and integrability by compensation.

Theorem [Wente 1969, Tartar 1983]

Let � and � be two functions in � ��

� � � � � � . Let � be the unique solution of

��� �

�� � �

� � � � � �� � � � � � � � � � � � � � � in � �

� � on � � � �

(54)

Then the following estimates hold true

� � � � � � � � � � � � �� � � � � � � � � � � � � � � � (55)

Recall that � , measurable, is in ���

� if and only if

��

��

� �� � � � s. t. � � � � � � �

��

��

� � � � � � �

Wente estimates and integrability by compensation. 1

Wente estimates and integrability by compensation.

Theorem [Wente 1969, Tartar 1983, Coifman-Lions-Meyer-Semmes 1989]

Let � and � be two functions in � ��

� � � � � � . Let � be the unique solution of

��� �

�� � �

� � � � � �� � � � � � � � � � � � � � � in � �

� � on � � � �

(56)

Then the following estimates hold true

� � � � � � � � � � � � �� � � ��

� � � � � � � � � � � � � � � � � � � (57)

Recall that � , measurable, is in ���

� if and only if

��

��

� �� � � � s. t. � � � � � � �

��

��

� � � � � � �

Regularity of � � � � Willmore surfaces. 1

Regularity of � � � Willmore surfaces.

Theorem 5 [R. 2006] � � �

� Willmore surfaces are analytic. �

Regularity of � � � � Willmore surfaces. 1

Regularity of � � � Willmore surfaces.

Theorem 5 [R. 2006] � � �

� Willmore surfaces are analytic. �

Proof :

Regularity of � � � � Willmore surfaces. 1

Regularity of � � � Willmore surfaces.

Theorem 5 [R. 2006] � � �

� Willmore surfaces are analytic. �

Proof :

� Recall

�� � � ��

� � � � �

� ,� � � ��

� and � � �� and

���

� � � ��

� � � � �� ��

� � �� � �

Regularity of � � � � Willmore surfaces. 1

Regularity of � � � Willmore surfaces.

Theorem 5 [R. 2006] � � �

� Willmore surfaces are analytic. �

Proof :

� Recall

�� � � ��

� � � � �

� ,� � � ��

� and � � �� and

���

� � � ��

� � � � �� ��

� � �� � �

� Hence

�� � ���

� therefore

��

� � �� � and � � � ���

� �

Regularity of � � � � Willmore surfaces. 1

Regularity of � � � Willmore surfaces.

Theorem 5 [R. 2006] � � �

� Willmore surfaces are analytic. �

Proof :

� Recall

�� � � ��

� � � � �

� ,� � � ��

� and � � �� and

���

� � � ��

� � � � �� ��

� � �� � �

� Hence

�� � ���

� therefore

��

� � �� � and � � � ���

� �

Recall that � , measurable, is in ���

� if and only if

� � �

� � � �

�� �

�� �� � � � s. t. � � � � � � �

��

��� � �

Further integrability by compensation. 1

Further integrability by compensation.

Further integrability by compensation. 1

Further integrability by compensation.

Corollary [Bethuel 1992]

Let � � � ��

� � � � � � and � � � ���

� � � � � � � . Let � solving of

��� �

�� � �

� � � � � �� � � � � � � � � � � � � � � in � �

� � on � � � �

(58)

Further integrability by compensation. 1

Further integrability by compensation.

Corollary [Bethuel 1992]

Let � � � ��

� � � � � � and � � � ���

� � � � � � � . Let � solving of

��� �

�� � �

� � � � � �� � � � � � � � � � � � � � � in � �

� � on � � � �

(59)

Then� � � � � � � � � � � � � � � � � � � � � � � (60)

Further integrability by compensation. 1

Further integrability by compensation.

Corollary [Bethuel 1992]

Let � � � ��

� � � � � � and � � � ���

� � � � � � � . Let � solving of

��� �

�� � �

� � � � � �� � � � � � � � � � � � � � � in � �

� � on � � � �

(61)

Then� � � � � � � � � � � � � � � � � � � � � � � (62)

Conclusion of the proof of the regularity :

Further integrability by compensation. 1

Further integrability by compensation.

Corollary [Bethuel 1992]

Let � � � ��

� � � � � � and � � � ���

� � � � � � � . Let � solving of

��� �

�� � �

� � � � � �� � � � � � � � � � � � � � � in � �

� � on � � � �

(63)

Then� � � � � � � � � � � � � � � � � � � � � � � (64)

Conclusion of the proof of the regularity : Applying this result to

��

� � ��

� � �� � � � � �� �

gives ��

� � ��

Further integrability by compensation. 1

Further integrability by compensation.

Corollary [Bethuel 1992]

Let � � � ��

� � � � � � and � � � ���

� � � � � � � . Let � solving of

��� �

�� � �

� � � � � �� � � � � � � � � � � � � � � in � �

� � on � � � �

(65)

Then� � � � � � � � � � � � � � � � � � � � � � � (66)

Conclusion of the proof of the regularity : Applying this result to

��

� � ��

� � �� � � � � �� �

gives ��

� � �� ... � � � � �� .

Further integrability by compensation. 1

Further integrability by compensation.

Corollary [Bethuel 1992]

Let � � � ��

� � � � � � and � � � ���

� � � � � � � . Let � solving of

��� �

�� � �

� � � � � �� � � � � � � � � � � � � � � in � �

� � on � � � �

(67)

Then� � � � � � � � � � � � � � � � � � � � � � � (68)

Conclusion of the proof of the regularity : Applying this result to

��

� � ��

� � �� � � � � �� �

gives ��

� � �� ... � � � � �� .

Then [Wente, Tartar, CLMS] applied to the red equation gives�� � � � �

� ....bootstraping gives� � � � ....

Point removability result for Willmore surfaces. 1

Point removability result for Willmore surfaces.

Point removability result for Willmore surfaces. 1

Point removability result for Willmore surfaces.

Theorem 6 [Kuwert-Schatzle ( � � � ) 2003, R. ( � � � )] Let

�� � � � � � � ,�

� � � � . Assumei)

�� is a Willmore immersion on � � � � � � ,

ii)

� � � �� �� � � � � � � � � �

iii)

� � � � � �

� � �

� � � � � �

� � �

� � �

where � � � ��

� � � � and � � � � � � ��� � � � s. t.

�� � � � �

Point removability result for Willmore surfaces. 1

Point removability result for Willmore surfaces.

Theorem 6 [Kuwert-Schatzle ( � � � ) 2003, R. ( � � � )] Let

�� � � � � � � ,�

� � � � . Assumei)

�� is a Willmore immersion on � � � � � � ,

ii)

� � � �� �� � � � � � � � � �

iii)

� � � � � �

� � �

� � � � � �

� � �

� � �

where � � � ��

� � � � and � � � � � � ��� � � � s. t.

�� � � � �

Then

�� � � � is a � ��

� submanifold of � � � � � � .

Point removability result for Willmore surfaces. 1

Point removability result for Willmore surfaces.

Theorem 6 [Kuwert-Schatzle ( � � � ) 2003, R. ( � � � )] Let

�� � � � � � � ,�

� � � � . Assumei)

�� is a Willmore immersion on � � � � � � ,

ii)

� � � �� �� � � � � � � � � �

iii)

� � � � � �

� � �

� � � � � �

� � �

� � �

where � � � ��

� � � � and � � � � � � ��� � � � s. t.

�� � � � �

Then

�� � � � is a � ��

� submanifold of � � � � � � . Moreover��

� � � � � s. t.

�� � � � � � � � � � � � � � ���

Point removability result for Willmore surfaces. 1

Point removability result for Willmore surfaces.

Theorem 6 [Kuwert-Schatzle ( � � � ) 2003, R. ( � � � )] Let

�� � � � � � � ,�

� � � � . Assumei)

�� is a Willmore immersion on � � � � � � ,

ii)

� � � �� �� � � � � � � � � �

iii)

� � � � � �

� � �

� � � � � �

� � �

� � �

where � � � ��

� � � � and � � � � � � ��� � � � s. t.

�� � � � �

Then

�� � � � is a � ��

� submanifold of � � � � � � . Moreover��

� � � � � s. t.

�� � � � � � � � � � � � � � ���

and

�� � � � ��

� � � � is an analytic Willmore surface. �

Point removability result for Willmore surfaces. 1

Point removability result for Willmore surfaces.

Theorem 6 [Kuwert-Schatzle ( � � � ) 2003, R. ( � � � )] Let

�� � � � � � � ,�

� � � � . Assumei)

�� is a Willmore immersion on � � � � � � ,

ii)

� � � �� �� � � � � � � � � �

iii)

� � � � � �

� � �

� � � � � �

� � �

� � �

where � � � ��

� � � � and � � � � � � ��� � � � s. t.

�� � � � �

Then

�� � � � is a � ��

� submanifold of � � � � � � . Moreover��

� � � � � s. t.

�� � � � � � � � � � � � � � ���

and

�� � � � ��

� � � � is an analytic Willmore surface. �

Remark : � � in iii) is optimal !

The Li-Yau � � condition. 1

The Li-Yau � � condition.

Theorem. [P.Li-S.T.Yau 1982]

The Li-Yau � � condition. 1

The Li-Yau � � condition.

Theorem. [P.Li-S.T.Yau 1982]

Let � be a closed 2-manifold. Let

�� � � � � � � be an immersion.

Assume� � � � � s.t.

��� � � � � � � � � � � � � � � � � � �

��

distinct.

The Li-Yau � � condition. 1

The Li-Yau � � condition.

Theorem. [P.Li-S.T.Yau 1982]

Let � be a closed 2-manifold. Let

�� � � � � � � be an immersion.

Assume� � � � � s.t.

��� � � � � � � � � � � � � � � � � � �

��

distinct.

Then�

��

� �� � � � � � � � � �

The Li-Yau � � condition. 1

The Li-Yau � � condition.

Theorem. [P.Li-S.T.Yau 1982]

Let � be a closed 2-manifold. Let

�� � � � � � � be an immersion.

Assume� � � � � s.t.

��� � � � � � � � � � � � � � � � � � �

��

distinct.

Then�

��

� �� � � � � � � � � �

In particular if

��

� �� � � � � � � � �

�� is an embedding. �

Weak compactness of Willmore surfaces below � � . 1

Weak compactness of Willmore surfaces below � � .

Theorem. [Kuwert-Schatzle ( � � � ) 2003, R. ( � � � )]

Weak compactness of Willmore surfaces below � � . 1

Weak compactness of Willmore surfaces below � � .

Theorem. [Kuwert-Schatzle ( � � � ) 2003, R. ( � � � )]

Let � � be a sequence of Willmore embeddings of closed surfaces in � � � . Assume

i)

Area � � � unif. bounded �

ii)

genus � � � unif. bounded �

iii)

� � � � �

��

��

� � �� � � � � � � � � �

Weak compactness of Willmore surfaces below � � . 1

Weak compactness of Willmore surfaces below � � .

Theorem. [Kuwert-Schatzle ( � � � ) 2003, R. ( � � � )]

Let � � be a sequence of Willmore embeddings of closed surfaces in � � � . Assume

i)

Area � � � unif. bounded �

ii)

genus � � � unif. bounded �

iii)

� � � � �

��

��

� � �� � � � � � � � � �

Then

� � � � s. t. � � � � � � � as current

If � �� � , � �

� � � is the integration along a smooth Willmore embedding � . �

Strong compactness, modulo the Mobius group action, of Willmore torii below � � . 1

Strong compactness, modulo the M obius group action, ofWillmore torii below � � .

Theorem. [Kuwert-Schatzle ( � � � ) 2003, R. ( � � � � � ) 2006]

Strong compactness, modulo the Mobius group action, of Willmore torii below � � . 1

Strong compactness, modulo the M obius group action, ofWillmore torii below � � .

Theorem. [Kuwert-Schatzle ( � � � ) 2003, R. ( � � � � � ) 2006]

Let � � � . Then

� � ��

��

embedded Willmore torii in � �

with � � � � � � � � �

��

is strongly compact modulo translations and conformal transformations. �

Strong compactness, modulo the Mobius group action, of Willmore torii below � � . 1

Strong compactness, modulo the M obius group action, ofWillmore torii below � � .

Theorem. [Kuwert-Schatzle ( � � � ) 2003, R. ( � � � � � ) 2006]

Let � � � . Then

� � ��

��

embedded Willmore torii in � �

with � � � � � � � � �

��

is strongly compact modulo translations and conformal transformations. �

The Proof is based on the point removability result and the following energy lowerbound obtained for non umbilic Willmore � � by the mean of geometrico-algebraicmethods ([Bryant (m=3) 1984, Montiel (m=4) 2000])

� � ��� �� � � � � � � � � �