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ABSTRACT. We add two sections to [8] and answer some questions asked there. In the first section we give another derivation of Theorem 1.1 of [8], which reveals the relation between the entropy formula, (1.4) of [8], and the well-known Li–Yau’s gradient estimate. As a by-product we obtain the sharp estimates on ‘Nash’s entropy ’ for manifolds with nonnegative Ricci curvature. We also show that the equality holds in Li–Yau’s gradient estimate, for some positive solution to the heat equation, at some positive time, implies that the complete Riemannian manifold with nonnegative Ricci curvature is isometric to R n. In the second section we derive a dual entropy formula which, to some degree, connects Hamilton’s entropy with Perelman’s entropy in the case of Riemann surfaces. 1. The relation with Li–Yau’s gradient estimates In this section we provide another derivation of Theorem 1.1 of [8] and discuss its relation with Li–Yau’s gradient estimates on positive solutions of heat equation. The formulation gives a sharp upper and lower bound estimates on Nash’s ‘entropy quantity ’ − � M H log Hdvin the case M has nonnegative Ricci curvature, where H is the fundamental solution (heat kernel) of the heat equation. This section is following the ideas in the Section 5 of [9]. Let u(x, t) be a positive solution to � ∂ ∂t − � � u(x, t) = 0 with

Year: 2004

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