Ends of Immersed Minimal and Willmore Surfaces in Asymptotically Flat Spaces

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.

arXiv commentThis second version corrects typos from the previous one and includes a more general hypothesis on the growth of the area of the complete surface