[CvGmt News] Seminari Calcolo delle Variazioni

[depascal at dm.unipi.it]

This is a memo for the talk of this afternoon

Wednesday October 21 
At 5 pm 
Aula Riunioni (Dipartimento di Matematica)

Dr. Reto Muller (Scuola Normale Superiore)

Title: Ricci flow coupled with harmonic map flow

Abstract: We investigate a new geometric flow which consists of a coupled system of the Ricci flow (with some additional terms) on a closed manifold M with the harmonic map flow of a map from this evolving manifold M to some closed target manifold N. This system can be interpreted as the gradient flow of an energy functional which is a modification of Perelman’s energy for the Ricci flow. Surprisingly, the coupled system may be less singular than the Ricci flow or the harmonic map flow alone. In particular, we can always rule out energy concentration of the map a-priori – without any assumptions on the curvature of the target manifold N – if some coupling parameter is chosen large enough. Moreover, it suffices to bound the curvature of (M,g(t)) to also obtain control on the map and all its derivatives. Besides these new phenomena, the flow shares many good properties with the Ricci flow. In particular, we can derive the monotonicity of an entropy functional similar to Perelman’s Ricci flow entropy and of so-called reduced volume functionals. These monotonicity results can be used to rule out non-trivial breathers and geometric collapsing at finite times.

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You find this news in the cvgmt preprint server: http://cvgmt.sns.it/news/20091021/