Ricci flow coupled with harmonic map flow
Ricci flow coupled with harmonic map flow
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DOI：
10.24033/asens.2161
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发表时间：
2009-12
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影响因子：
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通讯作者：
Reto Muller
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文献类型：
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作者：
Reto Muller
我们研究了一种新的几何流，它由封闭流形M上的Ricci流的耦合系统和从M到某个封闭目标流形N的映射的调和映射流组成，该映射流具有（可能与时间相关的）正耦合常数α。这个系统可以被解释为能量泛函F_alpha的梯度流，F_alpha是对雷奇流的佩雷尔曼能量F的修正，包括映射的狄利克雷能量。令人惊讶的是，耦合系统可能比单独的里奇流或谐波图流更不奇异。特别地，我们总是可以通过选择足够大的来排除先验的能量集中——不需要对目标流形N的曲率做任何假设。此外，如果有界远离0，它足以约束（M,g(t)）的曲率，从而也能控制和它的所有导数，这个结果显然对= 0是不成立的。除了这些新现象外，该流还具有许多与里奇流相同的良好性质。特别地，我们可以推导出熵泛函W_alpha的单调性，类似于Perelman的Ricci流熵W和所谓的缩小体积泛函。然后，我们应用这些单调性结果来排除有限时间内的非平凡呼吸和几何坍缩。
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.