Constrained Willmore Surfaces

Abstract

Constrained Willmore surfaces are conformal immersions of Riemann surfaces that are critical points of the Willmore energy $W=\int H^2$ under compactly supported infinitesimal conformal variations. Examples include all constant mean curvature surfaces in space forms. In this paper we investigate more generally the critical points of arbitrary geometric functionals on the space of immersions under the constraint that the admissible variations infinitesimally preserve the conformal structure. Besides constrained Willmore surfaces we discuss in some detail examples of constrained minimal and volume critical surfaces, the critical points of the area and enclosed volume functional under the conformal constraint.

17 pages, 8 figures; v2: Hopf tori added as an example, minor changes in presentation, numbering changed; v3: new abstract and appendix, several changes in presentation

References in corpus (2)

• A proof of the Willmore conjecture
• Soliton Spheres

Cited by in corpus (16)

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• Willmore Spheres in Compact Riemannian Manifolds
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• Computational p-Willmore Flow with Conformal Penalty
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• Constrained Willmore Tori in the 4--Sphere
• Darboux transforms and spectral curves of Hamiltonian stationary Lagrangian tori
• Polynomial conserved quantities for constrained Willmore surfaces
• Quartic differentials and harmonic maps in conformal surface geometry
• Nonlinear bending theories for non Euclidean plates
• Transformations of harmonic bundles and Willmore surfaces
• A characterization of the Ejiri torus in $S^5$
• On the Stability of the CMC Clifford Tori as Constrained Willmore Surfaces