Optimal transportation and monotonic quantities on evolving manifolds

Details Optimal transportation and monotonic quantities on evolving manifolds Optimal transportation and monotonic quantities on evolving manifolds

2009-08-23

arxiv.org/abs/0908.3293v2

Mathematics

Differential Geometry

8 pages, some corrections and extensions

Scientific paper

In this note we will adapt Topping's $\mathcal{L}$-optimal transportation theory for Ricci flow to a more general situation, i.e. to a closed manifold $(M,g_{ij}(t))$ evolving by $\partial_tg_{ij}=-2S_{ij}$, where $S_{ij}$ is a symmetric tensor field of (2,0)-type on $M$. We extend some recent results of Topping, Lott and Brendle, generalize the monotonicity of List's (and hence also of Perelman's) $\mathcal{W}$-entropy, and recover the monotonicity of M$\ddot{u}$ller's (and hence also of Perelman's) reduced volume.

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