几何流的单调体积公式
Monotone Volume Formulas for Geometric Flows
摘要：
我们考虑一个具有黎曼度量g（t）的闭流形M在方向-2S（t）上演化，其中S（t）是（M，g（t））上的对称双张量.我们证明了如果S满足一定的张量不等式，那么可以构造一个向前和向后的约化体积量，前者沿着流动不增加，后者沿着流动不减少。在S=Ric是M的里奇曲率的情况下，结果对应于Perelman著名的里奇流的约化体积单调性。本文的第二部分给出了一些其他的例子，主要的例子和动机是List的推广里奇流系统，里奇流与调和映射热流的耦合以及具有非负截面曲率的Lorentz流形中的中曲率流。我们的方法，我们发现这些流量的新的单调性公式。
英文摘要：
We consider a closed manifold M with a Riemannian metric g(t) evolving in direction -2S(t) where S(t) is a symmetric two-tensor on (M,g(t)). We prove that if S satisfies a certain tensor inequality, then one can construct a forwards and a backwards reduced volume quantity, the former being non-increasing, the latter being non-decreasing along the flow. In the case where S=Ric is the Ricci curvature of M, the result corresponds to Perelman's well-known reduced volume monotonicity for the Ricci flow. Some other examples are given in the second section of this article, the main examples and motivation for this work being List's extended Ricci flow system, the Ricci flow coupled with harmonic map heat flow and the mean curvature flow in Lorentzian manifolds with nonnegative sectional curvatures. With our approach, we find new monotonicity formulas for these flows.
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