$α$-curvatures and $α$-flows on low dimensional triangulated manifolds

Abstract

In this paper, we introduce two discrete curvature flows, which are called $α$-flows on two and three dimensional triangulated manifolds. For triangulated surface $M$, we introduce a new normalization of combinatorial Ricci flow (first introduced by Bennett Chow and Feng Luo \cite{CL1}), aiming at evolving $α$ order discrete Gauss curvature to a constant. When $αχ(M)\leq0$, we prove that the convergence of the flow is equivalent to the existence of constant $α$-curvature metric. We further get a necessary and sufficient combinatorial-topological-metric condition, which is a generalization of Thurston's combinatorial-topological condition, for the existence of constant $α$-curvature metric. For triangulated 3-manifolds, we generalize the combinatorial Yamabe functional and combinatorial Yamabe problem introduced by the authors in \cite{GX2,GX4} to $α$-order. We also study the $α$-order flow carefully, aiming at evolving $α$ order combinatorial scalar curvature, which is a generalization of Cooper and Rivin's combinatorial scalar curvature, to a constant.