Geometry of manifolds with densities

By Ovidiu Munteanu, Jiaping Wang. 2012

84 citations recorded by the source corpus.

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–Emery 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–Emery 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–Emery curvature is involved in our analysis.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/490e6ef6e60e2a9628794ad150b6fe45a57a6ad1

Citations in the corpus

• Stability and compactness for complete f -minimal surfaces (2012, unknown): “For instance, see the work of License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use Wei-Wylie [13], Munteanu-Wang [11, 12] and the references therein.”