Combinatorial Yamabe flow on infinitely triangulated hyperbolic surfaces

Abstract

We study the combinatorial Yamabe flow on infinitely triangulated surfaces with piecewise hyperbolic metrics. Under the assumptions of uniformly bounded vertex degree and $ε$-uniformly nondegenerate initial metric, we first establish the short-time existence of smooth solutions to the combinatorial Yamabe flow. Under the additional $ε$-uniformly Delaunay condition on the initial metric, we further obtain the short-time uniqueness of solutions to the flow. To address the potential degeneration of triangles along the evolution, we introduce an extended flow with generalized curvature, and establish the global existence of solutions to the extended flow. Furthermore, under uniformly bounded vertex degrees and some integrability condition, we establish the uniqueness of solutions to this extended flow, which follows from the stability property of the solutions. These results provide a well-posedness theory for both the hyperbolic combinatorial Yamabe flow (locally in time) and its extended flow (globally in time) on infinitely triangulated surfaces.