Analysis of weighted Laplacian and applications to Ricci solitons

Abstract

We study both function theoretic and spectral properties of the weighted Laplacian ${\Delta }_f$ on complete smooth metric measure space $(M,g,e^{-f}dv)$ with its Bakry-Émery curvature $Ri{c}_f$ 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 ${\Delta }_f$ in terms of the lower bound of $Ri{c}_f$ and the linear growth rate of $f.$ We also address the rigidity issue when the bottom spectrum achieves its optimal upper bound under a slightly stronger assumption that the gradient of $f$ is bounded. Applications to the study of the geometry and topology of gradient Ricci solitons are also considered. Among other things, it is shown that the volume of a noncompact shrinking Ricci soliton must be of at least linear growth. It is also shown that a nontrivial expanding Ricci soliton must be connected at infinity provided its scalar curvature satisfies a suitable lower bound.

Will appear in Comm. Anal. Geom

Cited by in corpus (4)

• Liouville theorems for $f$-harmonic maps into Hadamard spaces
• Volume growth, entropy and stability for translating solitons
• Liouville theorems, Volume growth, and volume comparison for Ricci shrinkers
• Local and global sharp gradient estimates for weighted $p$-harmonic functions