Refined position estimates for surfaces of Willmore type in Riemannian manifolds

TL;DR

This paper refines the position estimates of Willmore-type surfaces in Riemannian manifolds by analyzing small-area corrections and ambient curvature effects, leading to improved geometric understanding.

Contribution

It introduces new correction formulas for small-area surfaces and uses an adapted geometric center of mass to enhance position estimates based on ambient scalar curvature.

Findings

Derived correction terms for small-area Willmore surfaces

Established refined position estimates related to ambient scalar curvature

Connected geometric center of mass with curvature-based position adjustments

Abstract

In this paper we consider surfaces which are critical points of the Willmore functional subject to constrained area. In the case of small area we calculate the corrections to the intrinsic geometry induced by the ambient curvature. These estimates together with the choice of an adapted geometric center of mass lead to refined position estimates in relation to the scalar curvature of the ambient manifold.