Smooth metric measure spaces with non-negative curvature

By Ovidiu Munteanu, Jiaping Wang. 2011

153 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/8bbc7a51cc757c4891137434e2df786fe59db576

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In Applied Mathematics

1. Ovidiu Munteanu
2. Jiaping Wang
3. G. Carron
4. Jia-Yong Wu

In Geometry and Topology

1. Ovidiu Munteanu
2. Jiaping Wang
3. G. Carron
4. Jia-Yong Wu

Citations in the corpus

• Geometry of shrinking Ricci solitons (2014, unknown): “We adopt an argument in [20].”
• Analysis of weighted Laplacian and applications to Ricci solitons (2011, background): “Note that the conclusion here that the constant C is independent of x is stronger than that in [20].”