Horizontal Gauss Curvature Flow of Graphs in Carnot Groups

TL;DR

This paper establishes the existence of viscosity solutions for the horizontal Gauss curvature flow of graphs in Carnot groups, introducing a comparison principle for related degenerate parabolic equations.

Contribution

It provides the first proof of existence of solutions for this curvature flow in Carnot groups and develops a comparison principle for degenerate parabolic equations in this setting.

Findings

Existence of continuous viscosity solutions for the curvature flow.

Development of a comparison principle for degenerate parabolic equations in Carnot groups.

Extension of curvature flow analysis to sub-Riemannian geometries.

Abstract

We show the existence of continuous viscosity solutions to the equation describing the flow of a graph in the Carnot group G x R according to its horizontal Gauss curvature. In doing so, we prove a comparison principle for degenerate parabolic equations of the form u_t + F(D_0u, (D_0^2u)^*) = 0 for u defined on G.