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Analysis of weighted Laplacian and applications to Ricci solitons
Communications in analysis and geometry
We study both function theoretic and spectral properties of the weighted Laplacian Δ f on complete smooth metric measure space (M, g, e −f dv) with its Bakry-Émery curvature Ric f bounded from below by a constant. In particular, we establish a gradient estimate for positive f -harmonic functions and a sharp upper bound of the bottom spectrum of Δ f in terms of the lower bound of Ric f and the linear growth rate of f. We also address the rigidity issue when the bottom spectrum achieves itsdoi:10.4310/cag.2012.v20.n1.a3 fatcat:572zkbbhtfbtjhgjxakqrh2s5y