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.
The entropy formula for linear heat equation
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abstract
We derive the entropy formula for the linear heat equaiton on complete Riemannian manifolds with nonnegative Ricci curvature. As applications, we study the relation between the value of entropy and the volume of balls of various scales. The results are simpler version, without Ricci flow, of Perelman's recent results on volume non-collapsing.
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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.