Ricci flow coupled with harmonic map flow

By Reto Muller. 2009

155 citations recorded by the source corpus.

We investigate a new geometric flow which consists of 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 with a (possibly time-dependent) positive coupling constant alpha. This system can be interpreted as the gradient flow of an energy functional F_alpha which is a modification of Perelman's energy F for the Ricci flow, including the Dirichlet energy for the map phi. 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 - without any assumptions on the curvature of the target manifold N - by choosing alpha large enough. Moreover, if alpha is bounded away from zero it suffices to bound the curvature of (M,g(t)) to also obtain control of phi and all its derivatives - a result which is clearly not true for 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 entropy functional W_alpha similar to Perelman's Ricci flow entropy W and of so-called reduced volume functionals. We then apply these monotonicity results to rule out non-trivial breathers and geometric collapsing at finite times.

Subfields: Applied Mathematics, Geometry and Topology, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/1a1d54c9e410e35acb6a4dc5f27ea28c9c3779e8

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In Applied Mathematics

1. Anton Thalmaier
2. Robert Philipowski
3. Hongxin Guo
4. Feng-Yu Wang

In Geometry and Topology

1. Anton Thalmaier
2. Robert Philipowski
3. Hongxin Guo

In Radiology, Nuclear Medicine and Imaging

1. Anton Thalmaier
2. Robert Philipowski
3. Hongxin Guo

Citations in the corpus

• ENTROPY AND LOWEST EIGENVALUE ON EVOLVING MANIFOLDS (2013, methodology): “We thus present a unified formula of various W entropies established by various authors for different flows (including the static case), see [5, 8, 9, 11, 12, 13].”