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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.
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.
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.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.
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.
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]
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])
� � ��� �� � � � � � � � � �