Sharp gradient estimates and monotonicity in positive Ricci curvature

Cosmin Manea

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arXiv · 2512.05467

Sharp gradient estimates and monotonicity in positive Ricci curvature

Cosmin Manea

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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.

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Cosmin Manea. 2025-12-05. Sharp gradient estimates and monotonicity in positive Ricci curvature. https://arxiv.org/abs/2512.05467

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