Ends of Immersed Minimal and Willmore Surfaces in Asymptotically Flat Spaces

2015/08/06 by Yann Bernard, Bernard, Yann, Tristan Riviere +1
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #math.DG

paper · pdf · doi:10.48550/arxiv.1508.01391

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

arxiv created 2016/03/26 · arxiv updated 2016/03/29

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 L²-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.

Related

• Geometry of complete minimal surfaces at infinity and the Willmore index of their inversions
• Higher Regularity of Weak Limits of Willmore Immersions II
• On the asymptotic behavior of minimal surfaces in H²×R
• Foliations of asymptotically flat manifolds by surfaces of Willmore type
• Higher Regularity of Weak Limits of Willmore Immersions I