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Analysis Aspects of Willmore Surfaces ANALYSIS ASPECTS OF WILLMORE SURFACES. Tristan Riviere` Departement Mathematik ETH Z¨urich (e–mail: [email protected]) (Homepage: http://www.math.ethz.ch/ riviere) 3 juillet 2007, Paris. Des EDP au calcul scientifique. En l’honneur de Luc Tartar. £ 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 is bilinear symmetric from The - - - ¨ § © ¤ 2nd Fundamental form induced metric on the unit normal to oriented closed surface in ¡ ¡ . ¢ Curvatures for surfaces in £ ¤ ¤ ¤ ¥ . ¨ ( ¦ ¥ Gauss map : ¤ ¦ ¤ ¨ © § ¥ ¦ into ¨ ¦ ¡ . ¢ £ ). © ¦ ¤ the normal direction to ¨ © ¦ ¡ ¢ ¢ £ £ . ¨ © ¥ ¦ ¤ . (1) 1 Curvatures for surfaces in is bilinear symmetric from The - - - ¨ § © ¤ 2nd Fundamental form induced metric on the unit normal to oriented closed surface in ¡ ¡ . ¢ Curvatures for surfaces in £ ¤ ¤ ¤ ¥ . ¨ ( ¦ ¥ Gauss map : ¨ ¤ ¦ § ¤ ¦ ¨ © § ¥ ¦ into ¨ ¦ ¡ . ¢ ¡ ¢ £ £ ). ¥ ¤ © ¦ ¤ £ ¥ the normal direction to § ¦ ¨ ¨ © © ¦ ¡ ¢ ¢ £ £ . ¨ © ¥ ¦ ¤ . (3) (2) 1 Curvatures for surfaces in is bilinear symmetric from The - - - - - - - ¨ Gauss curvature Mean curvature vector Mean curvature Principal curvatures § © ¤ 2nd Fundamental form induced metric on the unit normal to oriented closed surface in ¡ ¡ . ¢ : Curvatures for surfaces in £ : ¤ ¤ ¤ : ¢ ¥ . ¨ ( ¦ : ¥ Gauss map : ¨ ¤ £ ¦ § ¤ ¤ ¤ ¨ ¦ ¨ £ ¨ © and § ¤ £ ¥ ¦ £ ¤ into ¨ ¦ ¡ . ¡ . £ ¢ ¡ ¢ ¨ £ £ £ ). ¥ © ¤ © ¦ . © ¤ . £ ¥ the normal direction to § ¦ ¨ ¨ © © ¦ ¡ ¢ ¢ £ £ . ¨ © ¥ ¦ ¤ . (5) (4) 1 The Willmore energy of a closed surface in Willmore energy The Willmore energy of a closed surface in : ¨ ¤ © ¡ ¡ . ¢ £ ¤ ¦ ¥ ¡ ¨ £ ¤ ¡ £ © ¢ £ ¤ § ¢ £ . (6) 1 The Willmore energy of a closed surface in Gauss-Bonnet Theorem Willmore energy The Willmore energy of a closed surface in : ¨ ¤ ¡ © ¢ : ¢ ¡ £ ¤ ¡ . ¢ £ ¡ ¤ ¢ ¨ ¤ © ¦ ¥ ¡ ¦ ¨ ¡ £ ¤ ¡ ¦ ¡ £ § © ¨ ¤ © ¢ £ § ¤ § ¢ £ . (8) (7) 1 The Willmore energy of a closed surface in Umbillic or anisotropy energy Gauss-Bonnet Theorem Willmore energy The Willmore energy of a closed surface in ¨ ¤ : ¨ © ¤ ¡ © ¢ ¡ : ¢ ¡ £ ¤ : ¡ . ¢ ¢ £ ¡ ¤ ¢ ¢ £ ¨ ¤ ¤ © ¦ ¥ ¦ ¥ ¡ ¦ ¨ ¡ £ ¡ ¤ ¨ £ ¡ ¦ ¤ ¡ £ § £ © ¨ ¤ © © ¢ £ § ¢ ¤ £ ¤ § ¢ £ § . (10) (11) (9) 1 The Willmore energy of a closed surface in Gauss equation Umbillic or anisotropy energy Gauss-Bonnet Theorem Willmore energy The Willmore energy of a closed surface in ¨ ¤ : : ¨ © ¤ ¡ © ¢ ¡ : ¢ ¡ £ ¤ : ¡ . ¢ ¢ ¢ £ ¡ ¤ ¢ ¢ £ ¨ ¤ ¤ ¥ ¨ © ¦ ¥ § ¦ ¥ ¡ ¦ ¨ ¡ £ ¡ ¤ ¨ ¡ ¨ £ ¡ © ¦ ¤ ¡ £ § £ © ¨ § ¤ © © ¢ £ § ¢ ¤ £ ¤ § ¢ £ § . (13) (15) (14) (12) 1 The Willmore energy of a closed surface in neous Conclusion Gauss equation Umbillic or anisotropy energy Gauss-Bonnet Theorem Willmore energy ¤ The Willmore energy of a closed surface in : Modulo a topological invariant norm of the Gauss map. ¨ ¤ : : ¨ © ¤ ¡ © ¢ ¡ : ¢ ¡ £ ¤ : ¡ . ¢ ¢ ¢ £ ¡ ¤ ¢ ¢ £ ¨ ¤ ¤ ¥ ¨ © ¦ ¥ § ¦ ¥ ¡ ¦ ¨ ¡ £ ¡ ¨ ¤ ¤ ¨ ¡ ¨ £ ¡ © © ¦ ¤ is comparable to the homge- ¡ £ § £ © ¨ § ¤ © © ¢ £ § ¢ ¤ £ ¤ § ¢ £ § . (17) (19) (18) (16) 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. Blaschke 1929. Willmore energy in various fields of science and technology. General Relativity Conformal geometry -...- Willmore 1965 . ¡ Hawking 1968 ¢ £ ¨ (presumably the origin). ¤ © ¨ ¥ . ¤ ¦ ¡ mass of 2 spheres : ¤ ¥ © § ¥ ¨ ¥ ¦ ¡ ¡ Thomsen, Schadow 1923 ¢ £ ¤ © 1 - Willmore energy in various fields of science and technology. vesicles and smectic A-liquid crystals. Blaschke 1929. Willmore energy in various fields of science and technology. General Relativity Cell Biology Conformal geometry . Helfrich 1973 -...- Willmore 1965 . ¡ Hawking 1968 ¢ £ ¨ (presumably the origin). ¤ © . Spontaneous curvature model for biomembranes, ¨ ¥ . ¤ ¦ ¡ mass of 2 spheres : ¤ ¥ © § ¥ ¨ ¥ ¦ ¡ ¡ Thomsen, Schadow 1923 ¢ £ ¤ © 1 - Willmore energy in various fields of science and technology. (Friesecke, James, 2001) M¨uller vesicles and smectic A-liquid crystals. Blaschke 1929. Willmore energy in various fields of science and technology. General Relativity Mechanic-Elasticity Cell Biology Conformal geometry . Helfrich 1973 -...- Willmore 1965 . ¡ Hawking 1968 . Non-linear plate theory. ¢ £ ¨ (presumably the origin). ¤ © . Spontaneous curvature model for biomembranes, ¨ . ¥ . ¤ ¦ ¡ mass of 2 spheres : ¤ ¥ © § ¥ ¨ ¥ ¦ ¡ -limit elastic energy ¡ Thomsen, Schadow 1923 ¢ £ ¤ © £ ¡ ¢ ¡ 1 - Willmore energy in various fields of science and technology. (Friesecke, James, 2001) M¨uller vesicles and smectic A-liquid crystals. Blaschke 1929. Willmore energy in various fields of science and technology. Optical design General Relativity Mechanic-Elasticity Cell Biology Conformal geometry . Helfrich 1973 . -...- Rubinstein 1990 Willmore 1965 . ¡ Hawking 1968 . Non-linear plate theory. ¢ £ ¨ (presumably the origin). ¤ © . Spontaneous curvature model for biomembranes, ¨ . ¥ . ¤ ¦ ¡ mass of 2 spheres : ¤ ¥ © § ¥ ¨ ¥ ¦ ¡ -limit elastic energy ¡ Thomsen, Schadow 1923 ¢ £ ¤ © £ ¡ ¢ ¡ 1 - Conformal invariance of Willmore energy. conformal diffeomorphism of Theorem . [Blaschke 1929] Conformal invariance of Willmore energy. Let ¥ ¦ ¨ ¤ ¡ be a closed oriented surface of ¨ ¢ ¤ £ © © ¤ then the following holds ¨ ¤ © § ¥ ¦ , let be a 1 Conformal invariance of Willmore energy. conformal diffeomorphism of Theorem ¨ © . [Blaschke 1929] ¡ Conformal invariance of Willmore energy. where ¢ ¡ Let ¢ ¥ ¦ ¨ ¤ ¡ be a closed oriented surface of £ ¨ ¢ ¢ ¤ £ © © ¤ then the following holds ¨ (inversion) ¤ © § ¥ ¦ , let be a 1 Generalization : Willmore energy of immersed surfaces in ¢ . Generalization : Willmore energy of immersed surfaces in . 1 Generalization : Willmore energy of immersed surfaces in Generalization : Willmore energy of immersed surfaces in - - - - - ¨ ¨ § ¡ © ¡ ¥ ¤ ¨ induced metric on £ smooth immersion of abstract oriented closed 2 dimensional manifold. : orthonormal projection onto the normal space given by © oriented space normal to the tangent 2-space at . in ¥ ¢ . ¨ © £ . ( Gauss Map ¢ ). 1 Generalization : Willmore energy of immersed surfaces in is bilinear symmetric. The Generalization : Willmore energy of immersed surfaces in - - - - - ¨ ¨ § ¡ © ¡ 2nd Fundamental form ¥ ¤ ¨ induced metric on £ smooth immersion of abstract oriented closed 2 dimensional manifold. : orthonormal projection onto the normal space given by © oriented ¡ ¢ £ ¤ ¥ space normal to the tangent 2-space at ¦ . : ¨ in § ¦ ¥ ¨ ¡ ¢ . ¢ £ © . ¡ ¥ ¤ ¨ ¡ ¨ ¡ ¢ £ © ¨ © § £ . ( Gauss Map ¢ (20) ). 1 Generalization : Willmore energy of immersed surfaces in is bilinear symmetric. The Generalization : Willmore energy of immersed surfaces in - - - - - - - ¨ ¨ Willmore energy Mean curvature vector § ¡ © ¡ 2nd Fundamental form ¥ ¤ ¨ induced metric on £ smooth immersion of abstract oriented closed 2 dimensional manifold. : orthonormal projection onto the normal space given by © oriented ¡ ¢ £ : ¤ ¥ ¨ space normal to the tangent 2-space at ¨ ¦ ¡ . : : © ¨ ¨ in § ¦ ¢ ¥ £ ¨ ¡ ¢ ¨ ¤ . ¢ £ ¡ ¨ © § . ¢ £ ¤ ¡ . ¥ ¤ ¨ ¡ ¨ ¡ ¢ £ © ¨ © § £ . ( Gauss Map ¢ (21) ). 1 Generalization : Willmore energy of immersed surfaces in is bilinear symmetric. in arbitrary dimension. In this talk we will restrict to The Generalization : Willmore energy of immersed surfaces in - - - - - - - ¨ ¨ Willmore energy Mean curvature vector § ¡ © ¡ 2nd Fundamental form ¥ ¤ ¨ induced metric on £ smooth immersion of abstract oriented closed 2 dimensional manifold. : orthonormal projection onto the normal space given by © oriented ¡ ¢ £ : ¤ ¥ ¨ space normal to the tangent 2-space at ¨ ¦ ¡ . : : © ¨ ¨ in § ¦ ¢ ¥ £ ¨ ¡ though most of the results presented are valid ¢ ¨ ¤ . ¢ £ ¡ ¨ © § . ¢ £ ¤ ¡ . ¥ ¤ ¨ ¡ ¨ ¡ ¢ £ © ¨ © § £ . ( Gauss Map ¢ (22) ). 1 Willmore Immersions. Willmore Immersions. 1 Willmore Immersions. following holds Definition : An immersion Willmore Immersions. ¨ ¡ : ¨ ¨ ¡ ¡ ¢ ¨ ¥ © ¦ ¤ ¥ ¦ is Willmore if § ¥ § ¨ ¤ ¡ £ ¢ ¨ ¢ ¥ ¦ © the ¨ 1 Willmore Immersions. following holds Definition - ¨ ¡ is a : An immersion minimal immersion Willmore Immersions. ¨ ¡ : : ¨ ¨ ¨ ¡ ¡ ¢ ¥ ¨ . ¥ © ¦ ¤ ¥ ¦ is Willmore if § ¥ § ¨ ¤ ¡ £ ¢ ¨ ¨ Examples : ¢ ¥ ¦ © the 1 Willmore Immersions. following holds Definition - - ¨ ¨ ¡ ¡ is a composition of a is a : An immersion minimal immersion Willmore Immersions. ¨ ¡ minimal immersion : : ¨ ¨ ¨ ¡ ¡ ¢ ¥ ¨ . ¥ © ¦ ¤ ¥ ¦ is Willmore if § and a ¥ § conformal transformation ¨ ¤ ¡ £ ¢ ¨ ¨ Examples : ¢ ¥ ¦ © the . 1 Willmore Immersions. following holds Definition - The - - ¨ ¨ holds for any closed surface ¡ ¡ is a composition of a is a round sphere : An immersion minimal immersion ¤ . Consequence of the following inequality Willmore Immersions. ¨ ¡ minimal immersion : ¨ : ¨ ¤ ¨ ¨ ¤ © ¡ with equality iff ¡ ¢ ¥ ¨ ¡ . ¥ © ¦ ¤ ¥ ¦ is Willmore if § ¢ and a £ ¥ ¤ ¤ is a round sphere. § conformal transformation ¦ ¡ ¨ ¤ ¡ £ ¢ ¨ ¨ Examples : ¢ ¥ ¦ © the . 1 Willmore Immersions. following holds Definition - The - The - - ¨ ¨ holds for any closed surface ¡ ¡ is a composition of a is a Willmore Torus round sphere : An immersion minimal immersion ¤ ¥ . Consequence of the following inequality
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