Smooth metric measure spaces with non-negative curvature

In this paper, we study both function theoretic and spectral properties oncomplete non-compact smooth metric measure space $(M,g, e^{-f}dv)$ withnon-negative Bakry–Émery Ricci curvature. Among other things, we derive agradient estimate for positive $f$-harmonic functions and obtain as aconsequence the strong Liouville property under the optimal sublinear growthassumption on $f.$ We also establish a sharp upper bound of the bottomspectrum 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 atinfinity, then $M$ must be a cylinder. As an application, we conclude steadyRicci solitons must be connected at infinity.

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Published 3 October 2011