Geometry of manifolds with densities

Ovidiu Munteanu, Jiaping Wang

Research output: Contribution to journalArticlepeer-review

22 Scopus citations

Abstract

We study the geometry of complete Riemannian manifolds endowed with a weighted measure, where the weight function is of quadratic growth. Assuming the associated Bakry-Émery curvature is bounded from below, we derive a new Laplacian comparison theorem and establish various sharp volume upper and lower bounds. We also obtain some splitting type results by analyzing the Busemann functions. In particular, we show that a complete manifold with nonnegative Bakry-Émery curvature must split off a line if it is not connected at infinity and its weighted volume entropy is of maximal value among linear growth weight functions.While some of our results are new even for the gradient Ricci solitons, the novelty here is that only a lower bound of the Bakry-Émery curvature is involved in our analysis.

Original languageEnglish (US)
Pages (from-to)269-305
Number of pages37
JournalAdvances in Mathematics
Volume259
DOIs
StatePublished - Jul 10 2014

Keywords

  • Bakry-Émery curvature
  • Splitting
  • Volume growth
  • Weighted Laplacian

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