-invariant homothetic solitons for inverse mean curvature flow in

Abstract

The inverse mean curvature flow (IMCF) has been extensively studied not only as a type of geometric flows, but also for its applications to geometric inequalities. The focus is primarily on homothetic solitons for the IMCF in this paper, which are special solutions deformed only homothetically under the flow. We completely classify the profile curves of the higher dimensional rotationally and birotationally symmetric homothetic solitons using the phase-plane analysis. As a characterization of the round cylinder, we obtain that any helicoidal homothetic soliton of IMCF must be round cylinders. In addition, we prove that any surface foliated by circles to be the homothetic soliton of IMCF, which is a cyclic homothetic soliton, is either a surface of revolution or a piece of a round sphere.