Combinatorial $p$-th Calabi flows on surfaces

TL;DR

This paper introduces a generalized combinatorial $p$-th Calabi flow on triangulated surfaces for all $p>1$, proving existence, convergence, and linking solutions to circle packing metrics with constant curvature.

Contribution

It extends the combinatorial Calabi flow from $p=2$ to any $p>1$, addressing nonlinearity challenges and establishing existence and convergence results.

Findings

Flow exists for all time for $p>1$

Flow converges iff a constant curvature circle packing metric exists

Generalizes previous $p=2$ results to all $p>1$

Abstract

For triangulated surfaces and any $p>1$, we introduce the combinatorial $p$-th Calabi flow which precisely equals the combinatorial Calabi flows first introduced in H. Ge's thesis when $p=2$. The difficulties for the generalizations come from the nonlinearity of the $p$-th flow equation when $p\neq 2$. Adopting different approaches, we show that the solution to the combinatorial $p$-th Calabi flow exists for all time and converges if and only if there exists a circle packing metric of constant (zero resp.) curvature in Euclidean (hyperbolic resp.) background geometry. Our results generalize the work of H. Ge, Ge-Xu and Ge-Hua on the combinatorial Calabi flow from $p=2$ to any $p>1$.