General Conformally Induced Mean Curvature Flow

TL;DR

This paper introduces a new mean curvature flow related to conformal Killing vector fields, extending isoperimetric inequality results to a broader class of hypersurfaces beyond traditional starshaped conditions.

Contribution

It is the first to analyze hypersurfaces starshaped with respect to general conformal Killing vector fields, broadening the scope of geometric flow applications.

Findings

Established isoperimetric inequalities for new classes of hypersurfaces.

Extended mean curvature flow techniques to non-traditionally starshaped hypersurfaces.

Generalized previous flows to include conformal Killing vector fields.

Abstract

This paper continues the investigation of isoperimetric inequalities through volume preserving and area decreasing mean curvature type flows related to conformal Killing vector fields. Results of this kind prior to this paper all studied convex hypersurfaces or hypersurfaces which are starshaped with respect to generalized dilations. This paper is the first to study results of this kind for hypersurfaces which are starshaped with respect to general conformal Killing vector fields perturbed by an isometric Killing vector field and our flows allow us to establish isoperimetric inequalities for a much wider class of hypersurfaces. For example, our results apply to hypersurfaces in $\mathbb{R}^{n+1}$ which are far from being starshaped in the traditional sense, but are starshaped with respect to the conformal Killing vector field composed of a dilation and a rotational vector field. The flow considered in this paper is a novel modification of the mean curvature-type flow first introduced by Guan and Li, which was later generalized by Guan-Li-Wang and Li-Pan.