TY - JOUR
T1 - Smooth metric measure spaces with non-negative curvature
AU - Munteanu, Ovidiu
AU - Wang, Jiaping
PY - 2011/7
Y1 - 2011/7
N2 - In this paper, we study both function theoretic and spectral properties on complete non-compact smooth metric measure space (M,g, e-f dv) with non-negative Bakry-Émery 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.
AB - In this paper, we study both function theoretic and spectral properties on complete non-compact smooth metric measure space (M,g, e-f dv) with non-negative Bakry-Émery 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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U2 - 10.4310/CAG.2011.v19.n3.a1
DO - 10.4310/CAG.2011.v19.n3.a1
M3 - Article
AN - SCOPUS:80053992719
SN - 1019-8385
VL - 19
SP - 451
EP - 486
JO - Communications in Analysis and Geometry
JF - Communications in Analysis and Geometry
IS - 3
ER -