Solitons for the inverse mean curvature flow

TL;DR

This paper classifies self-similar solutions to the inverse mean curvature flow, proving rigidity results for hyperspheres and expanding surfaces, and introduces new examples of rotational expanders with complex topologies.

Contribution

It provides a complete classification of planar solitons, proves the rigidity of hyperspheres among compact expanders, and constructs novel rotational expanders with intricate topologies.

Findings

All homothetic planar solitons are classified.

Hyperspheres are rigid among compact expanders.

New rotational expanders, including hypercylinders and infinite bottles, are constructed.

Abstract

We investigate self-similar solutions to the inverse mean curvature flow in Euclidean space. In the case of one dimensional planar solitons, we explicitly classify all homothetic solitons and translators. 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 big. 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.