On the evolution of convex hypersurfaces by the $Q_k$ flow

TL;DR

This paper establishes the existence, uniqueness, and regularity of solutions to the $Q_k$ flow for convex hypersurfaces, including the smoothing of interfaces with flat sides under certain conditions.

Contribution

It proves the existence and uniqueness of $C^{1,1}$ solutions to the $Q_k$ flow and demonstrates interface smoothing for convex hypersurfaces with flat sides.

Findings

Existence and uniqueness of $C^{1,1}$ solutions in the viscosity sense.

Smoothing of interfaces with flat sides under non-degeneracy conditions.

Short-time smooth evolution of the interface by the $Q_{k-1}$ flow.

Abstract

We prove the existence and uniqueness of a $C^{1,1}$ solution of the $Q_k$ flow in the viscosity sense for compact convex hypersurfaces $\Sigma_t$ embedded in $R^{n+1}$ ($n \geq 2$) . In particular, for compact convex hypersurfaces with flat sides we show that, under a certain non-degeneracy initial condition, the interface separating the flat from the strictly convex side, becomes smooth, and it moves by the $Q_{k-1}$ flow at least for a short time.