Ricci flow coupled with harmonic map flow
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 45 (2012) no. 1, p. 101-142

We investigate a coupled system of the Ricci flow on a closed manifold $M$ with the harmonic map flow of a map $\phi$ from $M$ to some closed target manifold $N$, $\frac{\partial }{\partial t}g=-2\mathrm{Rc}+2\alpha \nabla \phi \otimes \nabla \phi ,\phantom{\rule{2em}{0ex}}\frac{\partial }{\partial t}\phi ={\tau }_{g}\phi ,$ where $\alpha$ is a (possibly time-dependent) positive coupling constant. 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 $\phi$ a-priori by choosing $\alpha$ large enough. Moreover, it suffices to bound the curvature of $\left(M,g\left(t\right)\right)$ to also obtain control of $\phi$ and all its derivatives if $\alpha \ge \underline{\alpha }>0$. Besides these new phenomena, the flow shares many good properties with the Ricci flow. In particular, we can derive the monotonicity of an energy, an entropy and a reduced volume functional. We then apply these monotonicity results to rule out non-trivial breathers and geometric collapsing at finite times.

Nous étudions un système d’équations consistant en un couplage entre le flot de Ricci et le flot harmonique d’une fonction $\phi$ allant de $M$ dans une variété cible $N$, $\frac{\partial }{\partial t}g=-2\mathrm{Rc}+2\alpha \nabla \phi \otimes \nabla \phi ,\phantom{\rule{2em}{0ex}}\frac{\partial }{\partial t}\phi ={\tau }_{g}\phi ,$$\alpha$ est une constante de couplage strictement positive (et pouvant dépendre du temps). De manière surprenante, ce système couplé peut être moins singulier que le flot de Ricci ou le flot harmonique si ceux-ci sont considérés de manière isolée. En particulier, on peut toujours montrer que la fonction $\phi$ ne se concentre pas le long de ce système à condition de prendre $\alpha$ assez grand. De plus, il est suffisant de borner la courbure de $\left(M,g\left(t\right)\right)$ le long du flot pour obtenir le contrôle de $\phi$ et de toutes ses dérivées si $\alpha \ge \underline{\alpha }>0$. À part ces phénomènes nouveaux, ce flot possède certaines propriétés analogues à celles du flot de Ricci. En particulier, il est possible de montrer la monotonie d’une énergie, d'une entropie et d'une fonctionnelle volume réduit. On utilise la monotonie de ces quantités pour montrer l'absence de solutions en « accordéon » et l'absence d'effondrement en temps fini le long du flot.

DOI : https://doi.org/10.24033/asens.2161
Classification:  53C21,  53C43,  53C44,  58E20
Keywords: Ricci flow, harmonic map flow
@article{ASENS_2012_4_45_1_101_0,
author = {M\"uller, Reto},
title = {Ricci flow coupled with harmonic map flow},
journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
publisher = {Soci\'et\'e math\'ematique de France},
volume = {Ser. 4, 45},
number = {1},
year = {2012},
pages = {101-142},
doi = {10.24033/asens.2161},
zbl = {1247.53082},
mrnumber = {2961788},
language = {en},
url = {http://www.numdam.org/item/ASENS_2012_4_45_1_101_0}
}

Müller, Reto. Ricci flow coupled with harmonic map flow. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 45 (2012) no. 1, pp. 101-142. doi : 10.24033/asens.2161. http://www.numdam.org/item/ASENS_2012_4_45_1_101_0/

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