Journal
ADVANCES IN MATHEMATICS
Volume 259, Issue -, Pages 269-305Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.aim.2014.03.023
Keywords
Bakry-Emery curvature; Weighted Laplacian; Volume growth; Splitting
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Funding
- NSF [DMS-1262140, DMS-1105799]
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1105799] Funding Source: National Science Foundation
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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. (C) 2014 Elsevier Inc. All rights reserved.
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