Communications in Analysis and Geometry

Volume 28 (2020)

Number 1

Ends of immersed minimal and Willmore surfaces in asymptotically flat spaces

Pages: 1 – 57



Yann Bernard (School of Mathematical Sciences, Monash University, Melbourne, Victoria, Australia)

Tristan Rivière (Department of Mathematics, Eidgenössische Technische Hochschule (ETH) Zentrum, Zürich, Switzerland)


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 in any codimension.

Received 16 September 2016

Accepted 8 May 2017

Published 12 March 2020