On the anisotropic mean curvature flow in the Gaussian space

Abstract

In this paper, we study the anisotropic mean curvature flow in the Gaussian space endowed with a Finsler metric obtained from a Minkowski metric by a conformal deformation. We first derive the conformal transformation formulas for the fundamental tensor, the Cartan tensor, and the Chern connection of the conformally related Finsler metrics. Based on these formulas, we express the anisotropic mean curvature in the Gaussian space in terms of the corresponding anisotropic mean curvature with respect to the original Minkowski metric, and derive the associated evolution equation. Finally, we establish the short-time existence and uniqueness of smooth solutions by reformulating the geometric evolution equation as a strictly parabolic quasilinear partial differential equation.