Ends of Immersed Minimal and Willmore Surfaces in Asymptotically Flat Spaces

TL;DR

This paper analyzes the asymptotic behavior of ends of immersed Willmore and minimal surfaces in asymptotically flat spaces, revealing how the ambient metric's decay influences their geometric properties.

Contribution

It provides a detailed description of the asymptotics of ends of Willmore surfaces in asymptotically flat spaces, extending understanding beyond Euclidean settings.

Findings

Asymptotic behavior depends on ambient metric decay

Provides precise asymptotics for ends of Willmore surfaces

Results apply to minimal surfaces as a special case

Abstract

We study ends of an oriented, immersed, non-compact, complete Willmore surfaces, which are critical points of the integral of the square of the mean curvature, in asymptotically flat spaces of any dimension; assuming the surface has $L^2$-bounded second fundamental form and satisfies a weak power growth on the area. We give the precise asymptotic behavior of an end of such a surface. This asymptotic information is very much dependent on the way the ambient metric decays to the Euclidean one. Our results apply in particular to minimal surfaces.