Gaussian Mean curvature flow

TL;DR

This paper studies the evolution of convex hypersurfaces under mean curvature flow with exponential densities, providing a complete classification based on initial curvature bounds and density parameters.

Contribution

It offers a complete characterization of convex hypersurface evolution under density-modified mean curvature flow depending on initial curvature bounds and density parameters.

Findings

Classification of hypersurface evolution depending on density and initial curvature bounds

Explicit determination of the flow behavior for different parameter regimes

Insights into the geometric properties of hypersurfaces under weighted mean curvature flow

Abstract

We consider the evolution of a $n$-dimensional convex hypersurface in the euclidean space under mean curvature flow with densities $e^{\varepsilon \frac12 n\mu^2 |x|^2}$, $\varepsilon =\pm 1$, and completely determine it depending on the relation between $\mu$ and the upper or lower bound of the normal curvatures of the evolving hypersurface at time 0.