Academic literature on the topic 'Willmore functional'

For ann-dimensional hypersurface in unit sphere, we introduce an abstract Willmore type calledWn,F-Willmore functional, which generalizes the well-known classic Willmore functional. Its critical point is called theWn,F-Willmore hypersurface, for which the variational equation and Simons’ type integral equalities are obtained. Moreover, we construct a few examples ofWn,F-Willmore hypersurface and give a gap phenomenon characterization by use of our integral formula.

AbstractNoether’s theorem and the invariances of the Willmore functional are used to derive conservation laws that are satisfied by the critical points of the Willmore energy subject to generic constraints. We recover in particular previous results independently obtained by R. Capovilla and J. Guven, and by T. Rivière. Several examples are considered in detail.

AbstractWe discuss the minimum of Willmore functional of torus in a Riemannian manifold N, especially for the case that N is a product manifold. We show that when N = S2 × S1, the minimum of W(T2) is 0, and when N = R2 × S1, there exists no torus having least Willmore functional. When N = H2(−c) × S1, and x = γ × S1, the minimum of W(x) is $2\pi^2\sqrt{c}$.

Using techniques both of non linear analysis and geometric measure theory, we prove existence of minimizers and more generally of critical points for the Willmore functional and other $L^p$ curvature functionals for immersions in Riemannian manifolds. More precisely, given a $3$-dimensional Riemannian manifold $(M,g)$ and an immersion of a sphere $f:\Sp^2 \hookrightarrow (M,g)$ we study the following problems. 1) The Conformal Willmore functional in a perturbative setting: consider $(M,g)=(\Rtre,\eu+\epsilon h)$ the euclidean $3$-space endowed with a perturbed metric ($h=h_{\mu\nu}$ is a smooth field of symmetric bilinear forms); we prove, under assumptions on the trace free Ricci tensor and asymptotic flatness, existence of critical points for the Conformal Willmore functional $I(f):=\frac{1}{2}\int |A^\circ|^2 $ (where $A^\circ:=A-\frac{1}{2}H$ is the trace free second fundamental form). The functional is conformally invariant in curved spaces. We also establish a non existence result in general Riemannian manifolds. The technique is perturbative and relies on a Lyapunov-Schmidt reduction. \\ 2) The Willmore functional in a semi-perturbative setting: consider $(M,g)=(\Rtre, \eu+h)$ where $h=h_{\mu\nu}$ is a $C^{\infty}_0(\Rtre)$ field of symmetric bilinear forms with compact support and small $C^1$ norm. Under a general assumption on the scalar curvature we prove existence of a smooth immersion of $\Sp^2$ minimizing the Willmore functional $W(f):=\frac{1}{4} \int |H|^2$ (where $H$ is the mean curvature). The technique is more global and relies on the direct method in the calculus of variations. \\ 3) The functionals $E:=\frac{1}{2} \int |A|^2 $ and $W_1:=\int\left( \frac{|H|^2}{4}+1 \right)$ in compact ambient manifolds: consider $(M,g)$ a $3$-dimensional compact Riemannian manifold. We prove, under global conditions on the curvature of $(M,g)$, existence and regularity of an immersion of a sphere minimizing the functionals $E$ or $W_1$. The technique is global, uses geometric measure theory and regularity theory for higher order PDEs. \\ 4) The functionals $E_1:=\int \left( \frac{|A|^2}{2} +1 \right) $ and $W_1:=\int\left( \frac{|H|^2}{4}+1 \right)$ in noncompact ambient manifolds: consider $(M,g)$ a $3$-dimensional asymptotically euclidean non compact Riemannian $3$-manifold. We prove, under general conditions on the curvature of $(M,g)$, existence and regularity of an immersion of a sphere minimizing the functionals $E_1$ or $W_1$. The technique relies on the direct method in the calculus of variations. \\ 5) The supercritical functionals $\int |H|^p$ and $\int |A|^p$ in arbitrary dimension and codimension: consider $(N,g)$ a compact $n$-dimensional Riemannian manifold possibly with boundary. For any $2\leq m<n$ consider the functionals $\int |H|^p$ and $\int |A|^p$ with $p>m$, defined on the $m$-dimensional submanifolds of $N$. We prove, under assumptions on $(N,g)$, existence and partial regularity of a minimizer of such functionals in the framework of varifold theory. During the arguments we prove some new monotonicity formulas and new Isoperimetric Inequalities which are interesting by themselves.

Dans cette thèse on étudie la relaxée d'une fonctionnelle dépendant de la courbure moyenne des ensembles de niveau de la fonction. La relaxation est définie par rapport à la topologie forte de L^1 dans l'espace BV. En dimension deux, on peut exprimer la relaxée comme l'intégrale sur l'ensemble des niveaux de la fonction étudiée d'une énergie calculée sur un recouvrement des frontières essentielles d'ensembles de niveau par une famille limite de courbes. En dimension supérieure, on propose une nouvelle formulation pour le problème en définissant des varifolds associés aux mesures de Young-gradients, appelés Young varifolds. On se ramène ainsi à un problème de minimisation pour une fonctionnelle définie sur une certaine classe de Young varifolds. Grâce aux résultats précédents on peut montrer que cette formulation est appropriée en dimension deux. Toutefois, une caractérisation complète de la relaxée à l'aide des Young varifolds reste un problème ouvert.

This thesis is devoted to the study of elliptic equations. On the one hand, we study some qualitative properties, such as symmetry of solutions, on the other hand we explicitly construct some solutions vanishing near some fixed manifold. The main techniques are the moving planes method, in order to investigate the qualitative properties and the Lyapunov-Schmidt reduction.

En biologie, lorsqu'une quantité importante de phospholipides est insérée dans un milieu aqueux, ceux-Ci s'assemblent alors par paires pour former une bicouche, plus communément appelée vésicule. En 1973, Helfrich a proposé un modèle simple pour décrire la forme prise par une vésicule. Imposant la surface de la bicouche et le volume de fluide qu'elle contient, leur forme minimise une énergie élastique faisant intervenir des quantités géométriques comme la courbure, ainsi qu'une courbure spontanée mesurant l'asymétrie entre les deux couches. Les globules rouges sont des exemples de vésicules sur lesquels sont fixés un réseau de protéines jouant le rôle de squelette au sein de la membrane. Un des principaux travaux de la thèse fut d'introduire et étudier une condition de boule uniforme, notamment pour modéliser l'effet du squelette. Dans un premier temps, on cherche à minimiser l'énergie de Helfrich sans contrainte puis sous contrainte d'aire. Le cas d'une courbure spontanée nulle est connu sous le nom d'énergie de Willmore. Comme la sphère est un minimiseur global de l'énergie de Willmore, c'est un bon candidat pour être un minimiseur de l'énergie de Helfrich parmi les surfaces d'aire fixée. Notre première contribution dans cette thèse a été d'étudier son optimalité. On montre qu'en dehors d'un certain intervalle de paramètres, la sphère n'est plus un minimum global, ni même un minimum local. Par contre, elle est toujours un point critique. Ensuite, dans le cas de membranes à courbure spontanée négative, on se demande si la minimisation de l'énergie de Helfrich sous contrainte d'aire peut être effectuée en minimisant individuellement chaque terme. Cela nous conduit à minimiser la courbure moyenne totale sous contrainte d'aire et à déterminer si la sphère est la solution de ce problème. On montre que c'est le cas dans la classe des surfaces axisymétriques axiconvexes mais que ce n'est pas vrai en général.Enfin, lorsqu'une contrainte d'aire et de volume sont considérées simultanément, le minimiseur ne peut pas être une sphère qui n'est alors plus admissible. En utilisant le point de vue de l'optimisation de formes, la troisième et plus importante contribution de cette thèse est d'introduire une classe plus raisonnable de surfaces, pour laquelle l'existence d'un minimiseur suffisamment régulier est assurée pour des fonctionnelles et des contraintes générales faisant intervenir les propriétés d'ordre un et deux des surfaces. En s'inspirant de ce que fit Chenais en 1975 quand elle a considéré la propriété de cône uniforme, on considère les surfaces satisfaisant une condition de boule uniforme. On étudie d'abord des fonctionnelles purement géométriques puis nous autorisons la dépendance à travers la solution de problèmes aux limites elliptiques d'ordre deux posés sur le domaine intérieur à la surface<br>In biology, when a large amount of phospholipids is inserted in aqueous media, they immediatly gather in pairs to form bilayers also called vesicles. In 1973, Helfrich suggested a simple model to characterize the shapes of vesicles. Imposing the area of the bilayer and the volume of fluid it contains, their shape is minimizing a free-Bending energy involving geometric quantities like curvature, and also a spontanuous curvature measuring the asymmetry between the two layers. Red blood cells are typical examples of vesicles on which is fixed a network of proteins playing the role of a skeleton inside the membrane. One of the main work of this thesis is to introduce and study a uniform ball condition, in particular to model the effects of the skeleton. First, we minimize the Helfrich energy without constraint then with an area constraint. The case of zero spontaneous curvature is known as the Willmore energy. Since the sphere is the global minimizer of the Willmore energy, it is a good candidate to be a minimizer of the Helfrich energy among surfaces of prescribed area. Our first main contribution in this thesis was to study its optimality. We show that apart from a specific interval of parameters, the sphere is no more a global minimizer, neither a local minimizer. However, it is always a critical point. Then, in the specific case of membranes with negative spontaneous curvature, one can wonder whether the minimization of the Helfrich energy with an area constraint can be done by minimizing individually each term. This leads us to minimize total mean curvature with prescribed area and to determine if the sphere is a solution to this problem. We show that it is the case in the class of axisymmetric axiconvex surfaces but that it does not hold true in the general case. Finally, considering both area and volume constraints, the minimizer cannot be the sphere, which is no more admissible. Using the shape optimization point of view, the third main and most important contribution of this thesis is to introduce a more reasonable class of surfaces, in which the existence of an enough regular minimizer is ensured for general functionals and constraints involving the first- and second-Order geometric properties of surfaces. Inspired by what Chenais did in 1975 when she considered the uniform cone property, we consider surfaces satisfying a uniform ball condition. We first study purely geometric functionals then we allow a dependence through the solution of some second-Order elliptic boundary value problems posed on the inner domain enclosed by the shape

碩士<br>國立清華大學<br>數學系<br>94<br>In this thesis, we consider the Willmore functional both on surfaces and general Riemannian manifolds. In surfaces case, we study some basic properties of Willmore functional, for example, the relation between conformal area, and the first eigenvalue..., ect. In general case, we calculate the Euler-Lagrange Equation for Willmore functional.

by Lo Yiu Ming.<br>Thesis (M.Phil.)--Chinese University of Hong Kong, 1998.<br>Includes bibliographical references (leaves 89-90).<br>Abstract also in Chinese.<br>Abstract --- p.iii<br>Acknowledgements --- p.iv<br>Chapter Chapter 1. --- Introduction --- p.1<br>Chapter 1.1. --- Main Idea --- p.5<br>Chapter 1.2. --- Organization --- p.8<br>Chapter Chapter 2. --- Geometric and Analytic Preliminaries --- p.9<br>Chapter 2.1. --- A Review on Measure Theory --- p.9<br>Chapter 2.2. --- Submanifolds in Rn --- p.11<br>Chapter 2.3. --- Several Results from PDEs --- p.17<br>Chapter 2.4. --- Biharmonic Comparison Lemma --- p.20<br>Chapter Chapter 3. --- Approximate Graphical Decomposition --- p.24<br>Chapter 3.1. --- Some Preliminaries --- p.24<br>Chapter 3.2. --- Approximate Graphical Decomposition --- p.30<br>Chapter Chapter 4. --- Existence & Regularity of Measure-theoretic Limits of Minimizing Sequence --- p.41<br>Chapter 4.1. --- Willmore Functional and Area --- p.41<br>Chapter 4.2. --- Existence of Measure-theoretic Limit of Minimizing Sequence --- p.45<br>Chapter 4.3. --- Higher Regularity at Good Points --- p.54<br>Chapter 4.4. --- Convergence in Hausdorff Distance Sense --- p.62<br>Chapter 4.5. --- Regularity near Bad Points --- p.64<br>Chapter Chapter 5. --- Existence of Genus 1 Minimizers in Rn --- p.83<br>References --- p.89

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