Smooth metric measure spaces with non-negative curvature

Ovidiu Munteanu; Jiaping Wang

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arXiv · 1103.0746

Smooth metric measure spaces with non-negative curvature

Ovidiu Munteanu·Jiaping Wang

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Abstract

We study both function theoretic and spectral properties on complete noncompact smooth metric measure space $(M,g,e^{-f}dv)$ with nonnegative Bakry-\'{E}mery Ricci curvature. Among other things, we derive a gradient estimate for positive $f$-harmonic functions and obtain as a consequence the strong Liouville property under the optimal sublinear growth assumption on $f.$ We also establish a sharp upper bound of the bottom spectrum of the $f$-Laplacian in terms of the linear growth rate of $f.$ Moreover, we show that if equality holds and $M$ is not connected at infinity, then $M$ must be a cylinder. As an application, we conclude steady Ricci solitons must be connected at infinity.

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Ovidiu Munteanu, Jiaping Wang. 2011-03-03. Smooth metric measure spaces with non-negative curvature. https://arxiv.org/abs/1103.0746

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