Curvature functionals, p-Willmore energy, and the p-Willmore flow - PowerPoint PPT Presentation

Curvature functionals, p-Willmore energy, and the p-Willmore flow Eugenio Aulisa, Anthony Gruber, Magdalena Toda , Hung Tran magda.toda@ttu.edu Texas Tech University Magdalena Toda (Texas Tech University) p-Willmore energy; p-Willmore flow 1 / 40

Outline Introduction and Motivation 1 Variation of curvature functionals 2 The p-Willmore energy 3 The p-Willmore flow 4 Acknowledgements: (2) and (3) joint between the presenter, A. Gruber, and H. Tran. (4) joint between E. Aulisa and A. Gruber. Magdalena Toda (Texas Tech University) p-Willmore energy; p-Willmore flow 2 / 40

The Willmore energy Let R : M → R 3 be a smooth immersion of the closed surface M . Recall the Willmore energy functional � H 2 dS , W ( M ) = M where H is the mean curvature of the surface. Facts: Critical points of W ( M ) are called Willmore surfaces, and arise as natural generalizations of minimal surfaces. W ( M ) is invariant under reparametrizations, and less obviously under conformal transformations of the ambient metric (Mobius transformations of R 3 ). Magdalena Toda (Texas Tech University) p-Willmore energy; p-Willmore flow 3 / 40

The Willmore energy (2) From an aesthetic perspective, the Willmore energy produces surface fairing (i.e. smoothing). How to see this? 1 � � ( κ 1 − κ 2 ) 2 dS = ( H 2 − K ) dS = W ( M ) − 2 πχ ( M ) , 4 M M by the Gauss-Bonnet theorem. Conclusion: The Willmore energy punishes surfaces for being non-umbilic! Magdalena Toda (Texas Tech University) p-Willmore energy; p-Willmore flow 4 / 40

Examples of Willmore-type energies The Willmore energy arises frequently in mathematical biology, physics and computer vision – sometimes under different names. Helfrich-Canham energy, � k c (2 H + c 0 ) 2 + kK dS , E H ( M ) := M Bulk free energy density, � 2 k (2 H 2 − K ) dS , σ F ( M ) = M Surface torsion, � 4( H 2 − K ) dS S ( M ) = M When M is closed, all share critical surfaces with W ( M ). Magdalena Toda (Texas Tech University) p-Willmore energy; p-Willmore flow 5 / 40

General bending energy More generally, these energies are all special cases of a model for bending energy proposed by Sophie Germain in 1820, � B ( M ) = S ( κ 1 , κ 2 ) dS , M where S is a symmetric polynomial in κ 1 , κ 2 . By Newton’s theorem, this is equivalent to the functional � F ( M ) = E ( H , K ) dS , M where E is smooth in H = 1 2 ( κ 1 + κ 2 ) , K = κ 1 κ 2 . Conclusion: Studying F ( M ) is natural from the point of view of bending energy, and reveals similarities between examples of scientific relevance. Magdalena Toda (Texas Tech University) p-Willmore energy; p-Willmore flow 6 / 40

General bending energy (2) It is useful to study the functional F ( M ) on surfaces M ⊂ M 3 ( k 0 ) which are immersed in a 3-D space form of constant sectional curvature k 0 . Why leave Euclidean space? It’s mathematically relevant (e.g. conformal geometry in S 3 , geometry in the quaternions H ). Physicists care about immersions in “Minkowski space” which has constant sectional curvature − 1. Can be modeled as H 3 ∼ = { q ∈ H H | qq ∗ = 1 } (hyperbolic quaternions). The notion of bending energy differs depending on the ambient space! For example, ( κ 1 − κ 2 ) 2 = 4( H 2 − K + k 0 ). Particularly reasonable to study the variations of F ( M ), as they encode important geometric information about minimizing surfaces. Magdalena Toda (Texas Tech University) p-Willmore energy; p-Willmore flow 7 / 40

Framework for computing variations Consider a variation of the surface M , i.e. a 1-parameter family of compactly supported immersions r ( x , t ) as in the following diagram, M 3 ( k 0 ) � � F O ˜ r π r M 3 ( k 0 ) M × ( − ε, ε ) M 3 ( k 0 ) � � and a dual basis { ω I } such Choosing a local section { e J } of F O that ω I ( e J ) = δ I J , it follows that: g = ( ω 1 ) 2 + ( ω 2 ) 2 + ( ω 3 ) 2 . Metric on M 3 ( k 0 ) : Connection on M 3 ( k 0 ) : ∇ e I = e J ⊗ ω J I . Volume form on M 3 ( k 0 ) = ω 1 ∧ ω 2 ∧ ω 3 . Connection is Levi-Civita (torsion-free) when ω I J = − ω J I . Magdalena Toda (Texas Tech University) p-Willmore energy; p-Willmore flow 8 / 40

Framework for computing variations (2) The Cartan structure equations on M 3 ( k 0 ) are then d ω I = − ω I J ∧ ω J , J + 1 JKL ω K ∧ ω L . d ω I J = − ω I K ∧ ω K 2 R I We may assume the normal velocity of r satisfies ∂ r ∂ t = u N , for some smooth u : M × R → R . Pulling back the frame to M × R , we may further assume e 3 := N is normal to M × { t } for each t , in which case ω i = ω i ( i = 1 , 2) , ω 3 = u dt . Magdalena Toda (Texas Tech University) p-Willmore energy; p-Willmore flow 9 / 40

General first variation Using this, it is possible to compute the following necessary condition for criticality with respect to F . Theorem: Gruber, T., Tran The first variation of the curvature functional F is given by � δ E ( H , K ) dS M � 1 � � � � (2 H 2 − K + 2 k 0 ) E H + 2 HK E K − 2 H E = 2 E H + 2 H E K ∆ u + u M − E K � h , Hess u � dS , where E H , E K denote the partial derivatives of E with respect to H resp. K , and h is the shape operator of M (II = h N ). Magdalena Toda (Texas Tech University) p-Willmore energy; p-Willmore flow 10 / 40

General second variation Theorem: Gruber, T., Tran At a critical immersion of M , the second variation of F is given by � � � 1 � (∆ u ) 2 dS δ 2 4 E HH + 2 H E HK + 4 H 2 E KK + E K E ( H , K ) dS = M M � � E KK � h , Hess u � 2 dS − � � + E HK + 4 H E KK ∆ u � h , Hess u � dS M M � � � u �∇ K , ∇ u � − 3 u � h 2 , Hess u � − 2 h 2 ( ∇ u , ∇ u ) − | Hess u | 2 + E K dS M � � (2 H 2 − K + 2 k 0 ) E HH + 2 H (4 H 2 − K + 4 k 0 ) E HK + 8 H 2 K E KK + M � − 2 H E H + (3 k 0 − K ) E K − E u ∆ u dS � � (2 H 2 − K + 2 k 0 ) 2 E HH + 4 HK (2 H 2 − K + 2 k 0 ) E HK + 4 H 2 K 2 E KK + M � u 2 dS − 2 K ( K − 2 k 0 ) E K − 2 HK E H + 2( K − 2 k 0 ) E � 2 E H + 6 H E K − 2(2 H 2 − K + 2 k 0 ) E HK − 4 HK E KK � � + u � h , Hess u � dS M � � � � + E H + 4 H E K h ( ∇ u , ∇ u ) dS + E H u �∇ H , ∇ u � dS M M � |∇ u | 2 dS , − � 2( K − k 0 ) E K + H E H � M where the subscripts E HH , E HK , E KK denote the second partial derivatives of E in the appropriate variables. Magdalena Toda (Texas Tech University) p-Willmore energy; p-Willmore flow 11 / 40

Advantages of these variational results Valid in any space form of constant sectional curvature k 0 . Quantities involved are as elementary as possible; directly computable from surface fundamental forms. Can be used to studying many specific functionals. Example: these expressions immediately yield the known variation of the Willmore functional, � � � � H 2 dS = H ∆ u + 2 H ( H 2 − K + 2 k 0 ) u δ dS . M M It follows that closed Willmore surfaces in M 3 ( k 0 ) are characterized by the equation ∆ H + 2 H ( H 2 − K + 2 k 0 ) = 0 . Magdalena Toda (Texas Tech University) p-Willmore energy; p-Willmore flow 12 / 40

The p-Willmore energy It is further interesting to consider the p-Willmore energy , � H p dS , W p ( M ) = p ∈ Z ≥ 0 . M Notice that the Willmore energy is recovered as W 2 . Why generalize Willmore? Conformal invariance is beautiful but very un-physical: unnatural for bending energy. W 0 , W 1 , and W 2 are quite different. Are other W p different? We will see that the p-Willmore energy is highly connected to minimal surface theory when p > 2 !! Magdalena Toda (Texas Tech University) p-Willmore energy; p-Willmore flow 13 / 40

Variations of p-Willmore energy Corollary: Gruber, T., Tran The first variation of W p is given by � � � p 2 H p − 1 ∆ u + (2 H 2 − K + 2 k 0 ) pH p − 1 u − 2 H p +1 u � H p dS = δ dS , M M Moreover, the second variation of W p at a critical immersion is given by � � p ( p − 1) H p dS = H p − 2 (∆ u ) 2 dS δ 2 4 M M � pH p − 1 � h ( ∇ u , ∇ u ) + 2 u � h , Hess u � + u �∇ H , ∇ u � − H |∇ u | 2 � + dS M � � � (2 p 2 − 4 p − 1) H p − p ( p − 1) KH p − 2 + 2 p ( p − 1) k 0 H p − 2 + u ∆ u dS M � � 4 p ( p − 1) H p +2 − 2( p − 1)(2 p + 1) KH p + p ( p − 1) K 2 H p − 2 + M � + 4(2 p 2 − 2 p − 1) k 0 H p − 4 p ( p − 1) k 0 KH p − 2 + 4 p ( p − 1) k 2 u 2 dS . 0 H p − 2 Magdalena Toda (Texas Tech University) p-Willmore energy; p-Willmore flow 14 / 40

Connection to minimal surfaces In light of these variational results, define a p-Willmore surface to be any M satisfying the Euler-Lagrange equation, p 2∆ H p − 1 − p (2 H 2 − K + 2 k 0 ) H p − 1 + 2 H p +1 = 0 on M . Using integral estimates inspired by Bergner and Jakob [1], it is possible to show the following: Theorem: Gruber, T., Tran When p > 2, any p -Willmore surface M ⊂ R 3 satisfying H = 0 on ∂ M is minimal. More precisely, let p > 2 and R : M → R 3 be an immersion of the p-Willmore surface M with boundary ∂ M . If H = 0 on ∂ M , then H ≡ 0 everywhere on M . Magdalena Toda (Texas Tech University) p-Willmore energy; p-Willmore flow 15 / 40