Analysis of weighted Laplacian and applications to Ricci solitons

Ovidiu Munteanu, Jiaping Wang

Research output: Contribution to journalArticlepeer-review

93 Scopus citations


We study both function theoretic and spectral properties of the weighted Laplacian δf on complete smooth metric measure space (M,g, e?f dv) with its Bakry-Émery curvature Ricf 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 δf in terms of the lower bound of Ricf 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.

Original languageEnglish (US)
Pages (from-to)55-94
Number of pages40
JournalCommunications in Analysis and Geometry
Issue number1
StatePublished - Jan 2012


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