Gromov's volume growth conjecture is proven: Ric>=0 and Scal>=1 on a complete noncompact n-manifold force Vol(B(p,R))<=C_n R^(n-2) for all centers and radii.
Hamilton's gradient estimate for the heat kernel on complete manifolds
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abstract
In this paper we extend a gradient estimate of R. Hamilton for positive solutions to the heat equation on closed manifolds to bounded positive solutions on complete, non-compact manifolds with $Rc \geq -Kg$. We accomplish this extension via a maximum principle of L. Karp and P. Li and a Bernstein-type estimate on the gradient of the solution. An application of our result, together with the bounds of P. Li and S.T. Yau, yields an estimate on the gradient of the heat kernel for complete manifolds with non-negative Ricci curvature that is sharp in the order of $t$ for the heat kernel on ${\mathbb{R}}^n$.
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math.DG 1years
2026 1verdicts
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Heat kernel geometry and Gromov's volume growth conjecture
Gromov's volume growth conjecture is proven: Ric>=0 and Scal>=1 on a complete noncompact n-manifold force Vol(B(p,R))<=C_n R^(n-2) for all centers and radii.