On the deformation of discrete conformal factors on surfaces

TL;DR

This paper proves that the discrete Yamabe flow on surfaces can be extended to converge exponentially fast to a constant curvature metric without the need for surgeries, confirming Luo's conjecture.

Contribution

It establishes the convergence of the discrete Yamabe flow without surgeries, advancing understanding of discrete conformal geometry on surfaces.

Findings

Flow always extends to a convergent solution

Convergence is exponential

Confirms Luo's conjecture

Abstract

In \cite{Luo0}, Feng Luo conjectured that the discrete Yamabe flow will converge to the constant curvature PL-metric after finite number of surgeries on the triangulation. In this paper, we prove that the flow can always be extended (without surgeries) to a solution that converges exponentially fast to the constant curvature PL-metric.