Vol. 267, No. 1, 2014

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Eigenvalues and entropies under the harmonic-Ricci flow

Yi Li

Vol. 267 (2014), No. 1, 141–184

In this paper, the author discusses the eigenvalues and entropies under the harmonic-Ricci flow, which is the Ricci flow coupled with the harmonic map flow. We give an alternative proof of results for compact steady and expanding harmonic-Ricci breathers. In the second part, we derive some monotonicity formulas for eigenvalues of the Laplacian under the harmonic-Ricci flow. Finally, we obtain the first variation of the shrinker and expanding entropies of the harmonic-Ricci flow.

Eigenvalue, entropies, harmonic-Ricci flow, harmonic-Ricci breathers
Mathematical Subject Classification 2010
Primary: 53C44, 35K55
Received: 27 August 2012
Revised: 6 January 2013
Accepted: 16 January 2013
Published: 22 December 2013
Yi Li
Department of Mathematics
Johns Hopkins University
3400 N. Charles Street
Baltimore, MD 21218
United States
Department of Mathematics
Shanghai Jiao Tong University
800 Dong Chuan Road, Min Hang District
Shanghai, 200240