The evolution of complete non-compact graphs by powers of Gauss curvature

TL;DR

This paper investigates how convex complete non-compact graphs evolve over time when driven by positive powers of Gauss curvature, showing long-term evolution under certain convexity conditions even without initial differentiability.

Contribution

It establishes the long-term evolution of such graphs by positive powers of Gauss curvature, extending previous results to non-smooth initial data.

Findings

Graphs evolve by any positive power of Gauss curvature over time.

Initial local uniform convexity ensures evolution for all time.

The initial graph need not be differentiable.

Abstract

We study the evolution of convex complete non-compact graphs by positive powers of Gauss curvature. We show that if the initial complete graph has a local uniform convexity, then the graph evolves by any positive power of Gauss curvature for all time. In particular, the initial graph is not necessarily differentiable.