Mathematical Research Letters

Volume 19 (2012)

Number 1

Bounds on volume growth of geodesic balls under Ricci flow

Pages: 245 – 253



Qi S. Zhang (Department of Mathematics, University of California at Riverside, CA 92521, USA)


We prove a so-called $\kappa$ non-inflating property for Ricciflow, which provides an upper bound for volume ratio of geodesicballs over Euclidean ones, under an upper bound for scalarcurvature. This result can be regarded as the opposite statementof Perelman’s $\kappa$ non-collapsing property for Ricci flow.These two results together imply volume-doubling property forRicci flow without assuming Ricci curvature lower bound.

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