
Mathematics 2015
$α$curvatures and $α$flows on low dimensional triangulated manifoldsAbstract: In this paper, we introduce two discrete curvature flows, which are called $\alpha$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 $\alpha$ order discrete Gauss curvature to a constant. When $\alpha\chi(M)\leq0$, we prove that the convergence of the flow is equivalent to the existence of constant $\alpha$curvature metric. We further get a necessary and sufficient combinatorialtopologicalmetric condition, which is a generalization of Thurston's combinatorialtopological condition, for the existence of constant $\alpha$curvature metric. For triangulated 3manifolds, we generalize the combinatorial Yamabe functional and combinatorial Yamabe problem introduced by the authors in \cite{GX2,GX4} to $\alpha$order. We also study the $\alpha$order flow carefully, aiming at evolving $\alpha$ order combinatorial scalar curvature, which is a generalization of Cooper and Rivin's combinatorial scalar curvature, to a constant.
