Minimizers of the Willmore functional with a small area constraint

Abstract

We show the existence of a smooth spherical surface minimizing the Willmore functional subject to an area constraint in a compact Riemannian three-manifold, provided the area is small enough. Moreover, we classify complete surfaces of Willmore type with positive mean curvature in Riemannian three-manifolds.

Minor corrections

References in corpus (2)

• Willmore Spheres in Compact Riemannian Manifolds
• Willmore minimizers with prescribed isoperimetric ratio

Cited by in corpus (6)

• Lower-semicontinuity for the Helfrich problem
• Existence of generalized totally umbilic 2-spheres in perturbed 3-spheres
• Concentration of small Willmore spheres in Riemannian 3-manifolds
• Foliation by area-constrained Willmore spheres near a non-degenerate critical point of the scalar curvature
• Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds I: Minimization
• Some rigidity results for the Hawking mass and a lower bound for the Bartnik capacity