Analysis of weighted Laplacian and applications to Ricci solitons

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.

Full Text (PDF format)

Published 21 March 2012