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# Communications in Analysis and Geometry

## Volume 20 (2012)

### Number 1

### Analysis of weighted Laplacian and applications to Ricci solitons

Pages: 55 – 94

DOI: http://dx.doi.org/10.4310/CAG.2012.v20.n1.a3

#### Authors

#### Abstract

We study both function theoretic and spectral properties of the weightedLaplacian $\Delta_f$ on complete smooth metric measure space $(M,g,{\rm e}^{-f}dv)$with its Bakry–Émery curvature ${\rm Ric}_f$ bounded from below by a constant.In particular, we establish a gradient estimate for positive $f$-harmonicfunctions and a sharp upper bound of the bottom spectrum of $\Delta_f$ interms of the lower bound of ${\rm Ric}_{f}$ and the linear growth rate of $f.$ Wealso address the rigidity issue when the bottom spectrum achieves itsoptimal upper bound under a slightly stronger assumption that the gradientof $f$ is bounded.

Applications to the study of the geometry and topology of gradient Riccisolitons are also considered. Among other things, it is shown that thevolume of a noncompact shrinking Ricci soliton must be of at least lineargrowth. It is also shown that a nontrivial expanding Ricci soliton must beconnected at infinity provided its scalar curvature satisfies a suitablelower bound.