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In this paper, we introduce two discrete curvature flows, which are called (alpha )-flows on two and three dimensional triangulated manifolds. For a triangulated surface M, we introduce a new normalization of combinatorial Ricci flow [first introduced by Bennett Chow and Feng Luo (J Differ Geom 97–129, 2003)], aiming at evolving (alpha )-order discrete Gauss curvature to a constant. When (alpha chi (M)le 0), we prove that the convergence of the flow is equivalent to the existence of constant (alpha )-curvature metric. We further modify this flow to study the prescribing curvature problem on triangulated surfaces. For triangulated 3-manifolds, we generalize the combinatorial Yamabe functional and combinatorial Yamabe problem introduced by the authors Ge and Xu (Adv Math 267:470–497, 2014), (arXiv:1504.05814, 2015) to (alpha )-order. We also study the (alpha )-order flow, aiming at evolving (alpha )-order combinatorial scalar curvature, which is a generalization of Cooper and Rivin’s combinatorial scalar curvature, to a constant.

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