The $Q_k$ flow on complete non-compact graphs

TL;DR

This paper studies the evolution of complete non-compact graphs under the $Q_k$ flow, showing how the curvature flow progresses over time depending on the initial graph's enclosed sphere radius.

Contribution

It establishes the behavior of the $Q_k$ flow on complete non-compact graphs and determines the maximal existence time based on initial geometric conditions.

Findings

The $Q_k$ flow preserves completeness of the graph.

The maximal existence time depends on the initial enclosed sphere radius.

The flow evolves the graph according to the $Q_k$ curvature.

Abstract

We consider the $Q_k$ flow on complete non-compact graphs. We prove that a complete graph evolves by the $Q_k$ curvature up to some time $T$ depending on the radius of a sphere enclosed by the initial graph.