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Asymptotic estimates on the time derivative of entropy on a Riemannian manifold. (English) Zbl 1267.58017

In Perelman’s solution to the Poincaré conjecture, the W-functional (i.e. Perelman’s W-entropy) played an important role. Since then, there have been many attempts to understand or generalize this functional to other situations.
In this paper, the authors consider the entropy of the solution to the heat equation on a Riemannian manifold and aim to estimate the asymptotic behavior of the time derivative of the entropy, which is motivated by the early work of Lei Ni, where he presented the expression for the time derivative of the entropy and studied its properties. When the manifold is compact, the authors provide an estimate on the rate of change of the entropy in terms of the lower bound on the Ricci curvature, and also give an estimate on the rate of change of entropy in terms of the spectral gap, which implies that the time derivative of the entropy always has an exponential decay, even though Ricci is bounded below by a negative constant Finally, based on the explicit computation for the three dimensional hyperbolic space, the authors also show that the time derivative of the entropy is asymptotically bounded by two positive constants.
Note that the compactness of the manifold is only used to guarantee that the entropy is finite and the integration by parts can be applied, thus it is a natural problem to ask if the same argument works as well for non-compact manifolds, which is interesting and valuable.

MSC:

58J35 Heat and other parabolic equation methods for PDEs on manifolds
58J05 Elliptic equations on manifolds, general theory
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