Discrete Yamabe flows with R-curvature revisited

Abstract

Using Ge-Jiang-Shen's extension method [8], we extend Ge-Xu's discrete Yamabe flow with R-curvature [11]. We show the solution to the extended flow is unique. We give a picture of the singularities in finite time, that the degeneration is at least codimension two. We show the solution exists for all time t≥0 if all vertex degree are no less than 23, which partly confirm the Conjecture 1 in [11]. We also give two sufficient conditions (the “energy gap” condition and the “big regular triangulation” condition) for the sub-convergence to a constant R-curvature metric.