Translating Solitons for the Inverse Mean Curvature Flow

Abstract

In this paper, we investigate translating solitons for the inverse mean curvature flow (IMCF), which is a special solution deformed only for translation under the flow. The IMCF has been studied extensively not only as a type of a natural geometric flow, but also for obtaining various interesting geometric inequalities. We show that the translating solitons that are either ruled surfaces or translation surfaces are cycloid cylinders, and completely classify 2-dimensional helicoidal translating solitons and the higher dimensional rotationally symmetric translating solitons using the phase-plane analysis. The surface foliated by circles, which is called a cyclic surface, is regarded in terms of being the translating soliton for the IMCF, and then it is a surface of revolution whose revolution axis is parallel to the translating direction. In particular, we extend the result to a higher dimension, namely, the n-dimensional translating soliton foliated by spheres lying on parallel hyperplanes in $$\mathbb {R}^{n+1}$$ must be a rotationally symmetric hypersurface whose rotation axis is parallel to the translating direction.