Solitons for the inverse mean curvature flow

Gregory Drugan, Hojoo Lee and Glen Wheeler

Vol. 284 (2016), No. 2, 309–326

DOI: 10.2140/pjm.2016.284.309

Abstract

We investigate self-similar solutions to the inverse mean curvature flow in Euclidean space. Generalizing Andrews’ theorem that circles are the only compact homothetic planar solitons, we apply the Hsiung–Minkowski integral formula to prove the rigidity of the hypersphere in the class of compact expanders of codimension one. We also establish that the moduli space of compact expanding surfaces of codimension two is large. Finally, we update the list of Huisken–Ilmanen’s rotational expanders by constructing new examples of complete expanders with rotational symmetry, including topological hypercylinders, called infinite bottles, that interpolate between two concentric round hypercylinders.

Keywords

inverse mean curvature flow, self-similar solutions

Mathematical Subject Classification 2010

Primary: 53C44

Milestones

Received: 25 October 2015

Revised: 26 April 2016

Accepted: 27 April 2016

Published: 30 August 2016