Minimizers of the Willmore functional with a small area constraint

TL;DR

This paper proves the existence of smooth spherical minimizers of the Willmore functional under small area constraints in Riemannian three-manifolds and classifies certain complete surfaces with positive mean curvature.

Contribution

It establishes the existence of area-constrained Willmore minimizers and classifies specific Willmore surfaces in Riemannian three-manifolds, advancing geometric analysis.

Findings

Existence of smooth spherical minimizers for small area constraints.

Classification of complete Willmore surfaces with positive mean curvature.

Results applicable to compact Riemannian three-manifolds.

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.