Sharp gradient estimates and monotonicity in positive Ricci curvature

Abstract

We prove a sharp gradient estimate for the natural Green's function of a closed manifold with positive Ricci curvature. We also show that this estimate is closely related to a family of monotonicity formulae. These results extend those previously obtained by Colding and Minicozzi for open manifolds with non-negative Ricci curvature. We further obtain several geometric applications, including a new proof of Bishop's volume comparison theorem in dimension four.

42 pages. Corrected various typos, added a few references, and revised Remark 6.4 and the proof of Proposition B.3, based on the referee's comments; the main results are unchanged. Final version, accepted for publication in Crelle's Journal