Ricci flow coupled with harmonic map flow

Reto Müller

Annales scientifiques de l'École Normale Supérieure (2012)

  • Volume: 45, Issue: 1, page 101-142
  • ISSN: 0012-9593

Abstract

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We investigate a coupled system of the Ricci flow on a closed manifold M with the harmonic map flow of a map φ from M to some closed target manifold N , ∂ ∂ t g = - 2 Rc + 2 α ∇ φ ⊗ ∇ φ , ∂ ∂ t φ = τ g φ , where α 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  φ a-priori by choosing α large enough. Moreover, it suffices to bound the curvature of  ( M , g ( t ) ) to also obtain control of  φ and all its derivatives if α ≥ α ̲ > 0 . Besides these new phenomena, the flow shares many good properties with the Ricci flow. In particular, we can derive the monotonicity of anenergy, 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.

How to cite

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Müller, Reto. "Ricci flow coupled with harmonic map flow." Annales scientifiques de l'École Normale Supérieure 45.1 (2012): 101-142. <http://eudml.org/doc/272187>.

@article{Müller2012,
abstract = {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 $N$,\[ \frac\{\partial \}\{\partial t\} g = -2\mathrm \{Rc\}\{\} + 2\alpha \nabla \phi \otimes \nabla \phi ,\qquad \frac\{\partial \}\{\partial t\} \phi = \tau \_g \phi , \]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 $(M,g(t))$ to also obtain control of $\phi $ and all its derivatives if $\alpha \ge \underline\{\alpha \}&gt;0$. Besides these new phenomena, the flow shares many good properties with the Ricci flow. In particular, we can derive the monotonicity of anenergy, 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.},
author = {Müller, Reto},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {Ricci flow; harmonic map flow},
language = {eng},
number = {1},
pages = {101-142},
publisher = {Société mathématique de France},
title = {Ricci flow coupled with harmonic map flow},
url = {http://eudml.org/doc/272187},
volume = {45},
year = {2012},
}

TY - JOUR
AU - Müller, Reto
TI - Ricci flow coupled with harmonic map flow
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2012
PB - Société mathématique de France
VL - 45
IS - 1
SP - 101
EP - 142
AB - 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 $N$,\[ \frac{\partial }{\partial t} g = -2\mathrm {Rc}{} + 2\alpha \nabla \phi \otimes \nabla \phi ,\qquad \frac{\partial }{\partial t} \phi = \tau _g \phi , \]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 $(M,g(t))$ to also obtain control of $\phi $ and all its derivatives if $\alpha \ge \underline{\alpha }&gt;0$. Besides these new phenomena, the flow shares many good properties with the Ricci flow. In particular, we can derive the monotonicity of anenergy, 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.
LA - eng
KW - Ricci flow; harmonic map flow
UR - http://eudml.org/doc/272187
ER -

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