Second boundary value problem for the Hessian curvature flow

TL;DR

This paper studies the evolution and long-term behavior of convex hypersurfaces under the $k$-Hessian curvature flow with second boundary conditions, extending previous results from Gauss curvature flow.

Contribution

It extends the analysis of boundary value problems from Gauss curvature flow to the more general $k$-Hessian curvature flow, establishing existence, convergence, and boundary estimates.

Findings

Flow exists for all time and converges to a translating solution.

Orthogonal invariance technique helps obtain boundary $C^2$ estimates.

Generalizes previous results from Gauss curvature to $k$-Hessian curvature flow.

Abstract

We investigate the evolution of strictly convex hypersurfaces driven by the $k$-Hessian curvature flow, subject to the second boundary condition. We first explore the translating solutions corresponding to this boundary value problem. Next, we establish the long-time existence of the flow and prove that it converges to a translating solution. To overcome the difficulty of driving boundary $C^2$ estimates, we employ an orthogonal invariance technique. Using this method, we extend the results of Schn\"urer-Smoczyk \cite{Schnurer2003} and Schn\"urer \cite{Schnurer2002} from the second boundary value problem of Gauss curvature flow to $k$-Hessian curvature flow.