The scalar curvature flow with a flat side

TL;DR

This paper analyzes the behavior of a convex scalar curvature flow with a flat side, establishing interface propagation, curvature estimates, and regularity results, leading to short- and long-term smooth solutions.

Contribution

It provides new insights into the interface dynamics and regularity of scalar curvature flows with flat sides, including optimal estimates and existence results.

Findings

Interface propagates with finite speed until flat side vanishes

Established optimal decay and derivative estimates near the interface

Proved short-time and all-time existence of smooth solutions

Abstract

We study the near-the-interface behavior of a compact convex scalar curvature flow with a flat side. Under suitable initial conditions on the flat side, we show that the interface propagates with a finite and non-degenerate speed until the flat side vanishes. Then we get optimal derivative estimates of the pressure-like function, optimal decay estimates of curvatures near the interface, and an Aronson-B\'enilan-type curvature lower bound, from which we obtain the H\"older regularity of the ratio of the curvature to the optimal decay rate up to the free boundary. In the end, we obtain the short-time and all-time existence of the solution, smooth up to the interface.