Embedded surfaces of arbitrary genus minimizing the Willmore energy under isoperimetric constraint

Abstract

The isoperimetric ratio of an embedded surface in $R^3$ is defined as the ratio of the area of the surface to power three to the squared enclosed volume. The aim of the present work is to study the minimization of the Willmore energy under fixed isoperimetric ratio when the underlying abstract surface has fixed genus $g\ge 0$. The corresponding problem in the case of spherical surfaces, i.e. $g=0$, was recently solved by Schygulla with different methods.

38 pages

References in corpus (3)

• Willmore Spheres in Compact Riemannian Manifolds
• Willmore minimizers with prescribed isoperimetric ratio
• Immersed Spheres of Finite Total Curvature into Manifolds

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• Phase field models for thin elastic structures with topological constraint
• Elastic curves and phase transitions
• Li-Yau inequalities for the Helfrich functional and applications
• A Resolution of the Poisson Problem for Elastic Plates
• Embedded Delaunay tori and their Willmore energy
• A strict inequality for the minimisation of the Willmore functional under isoperimetric constraint
• Lower-semicontinuity for the Helfrich problem
• The Willmore flow with prescribed isoperimetric ratio
• Foliation by area-constrained Willmore spheres near a non-degenerate critical point of the scalar curvature