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Minimizers of the sharp log entropy on manifolds with non-negative Ricci curvature and flatness. (English) Zbl 1408.53092

Summary: Consider the scaling invariant, sharp log entropy (functional) introduced by F. B. Weissler [Trans. Am. Math. Soc. 237, 255–269 (1978; Zbl 0376.47019)] on noncompact manifolds with nonnegative Ricci curvature. It can also be regarded as a sharpened version of G. Perelman’s W entropy in the stationary case [arXiv e-print service, Paper No. 0211159, 39 p. (2002; Zbl 1130.53001)]. We prove that it has a minimizer if and only if the manifold is isometric to \(\mathbb{R}^n\). Using this result, it is proven that a class of noncompact manifolds with nonnegative Ricci curvature is isometric to \(\mathbb{R}^n\). Comparing with the well known flatness results in [M. T. Anderson, J. Am. Math. Soc. 2, No. 3, 455–490 (1989; Zbl 0694.53045); R. Bartnik, Commun. Pure Appl. Math. 39, 661–693 (1986; Zbl 0598.53045); S. Bando et al., Invent. Math. 97, No. 2, 313–349 (1989; Zbl 0682.53045)] on asymptotically flat manifolds and asymptotically locally Euclidean (ALE) manifolds, their decay or integral condition on the curvature tensor is replaced by the condition that the metric converges to the Euclidean one in \(C^1\) sense at infinity. No second order condition on the metric is needed.

MSC:

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)

Keywords:

decay
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