arXiv Analytics

Sign in

arXiv:1011.2728 [math.DG]AbstractReferencesReviewsResources

The energy of a smooth metric measure space and applications

Jeffrey S. Case

Published 2010-11-11, updated 2012-05-03Version 3

We introduce and study the notion of the energy of a smooth metric measure space, which includes as special cases the Yamabe constant and Perelman's $\nu$-entropy. We then investigate some properties the energy shares with these constants, in particular its relationship with the $\kappa$-noncollapsing property. Finally, we use the energy to prove a precompactness theorem for the space of compact quasi-Einstein smooth metric measure spaces, in the spirit of similar results for Einstein metrics and gradient Ricci solitons.

Comments: 44 pages; rewritten to use the more standard language of smooth metric measure spaces, and corrected some errors in the appendix
Categories: math.DG
Subjects: 53C21, 53C25
Related articles: Most relevant | Search more
arXiv:0712.1398 [math.DG] (Published 2007-12-10, updated 2008-06-05)
Prolongations of Lie algebras and applications
arXiv:1308.2263 [math.DG] (Published 2013-08-10, updated 2016-05-20)
Algebraic topology of $G_2$ manifolds
arXiv:math/0203138 [math.DG] (Published 2002-03-14, updated 2002-09-06)
Some properties of the Schouten tensor and applications to conformal geometry