Small surfaces of Willmore type in Riemannian manifolds

TL;DR

This paper studies small Willmore-type surfaces in Riemannian manifolds, showing their existence relates to scalar curvature critical points and strengthening non-existence results in regions with non-zero scalar curvature.

Contribution

It establishes a link between small Willmore surfaces and scalar curvature critical points, providing new non-existence results in Riemannian geometry.

Findings

Existence of small Willmore surfaces implies scalar curvature has a critical point at the center.

Strengthened non-existence results for Willmore critical points where scalar curvature is non-zero.

Provides estimates connecting surface properties with scalar curvature behavior.

Abstract

In this paper we investigate the properties of small surfaces of Willmore type in Riemannian manifolds. By \emph{small} surfaces we mean topological spheres contained in a geodesic ball of small enough radius. In particular, we show that if there exist such surfaces with positive mean curvature in the geodesic ball $B_r(p)$ for arbitrarily small radius $r$ around a point $p$ in the Riemannian manifold, then the scalar curvature must have a critical point at $p$. As a byproduct of our estimates we obtain a strengthened version of the non-existence result of Mondino \cite{Mondino:2008} that implies the non-existence of certain critical points of the Willmore functional in regions where the scalar curvature is non-zero.