Monotone volume formulas for geometric flows
File(s)
Published version
Author(s)
Müller, R
Type
Journal Article
Abstract
We consider a closed manifold M with a Riemannian metric g(t) evolving in direction -2S(t) where S(t) is a symmetric two-tensor on (M,g(t)). We prove that if S satisfies a certain tensor inequality, then one can construct a forwards and a backwards reduced volume quantity, the former being non-increasing, the latter being non-decreasing along the flow. In the case where S=Ric is the Ricci curvature of M, the result corresponds to Perelman's well-known reduced volume monotonicity for the Ricci flow. Some other examples are given in the second section of this article, the main examples and motivation for this work being List's extended Ricci flow system, the Ricci flow coupled with harmonic map heat flow and the mean curvature flow in Lorentzian manifolds with nonnegative sectional curvatures. With our approach, we find new monotonicity formulas for these flows.
Date Issued
2010-06
Citation
Journal fuer die Reine und Angewandte Mathematik: Crelle's journal, 2010, 2010 (643), pp.39-57
ISSN
0075-4102
Publisher
WALTER DE GRUYTER & CO
Start Page
39
End Page
57
Journal / Book Title
Journal fuer die Reine und Angewandte Mathematik: Crelle's journal
Volume
2010
Issue
643
Copyright Statement
© Walter de Gruyter 2010
Description
18.02.13 KB. Publisher permits published version to be added to Spiral. Gruyter
Identifier
http://arxiv.org/abs/0905.2328
Subjects
Geometric flows
Publication Status
Published