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**1 - 10**of**10**### Harnack estimates for ricci flow on a warped product. preprint arXiv:1211.6448

, 2012

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### OPTIMAL TRANSPORTATION AND MONOTONIC QUANTITIES ON EVOLVING MANIFOLDS

, 908

"... Abstract. In this note we will adapt Topping’s L-optimal transportation theory for Ricci flow to a more general situation, i.e. to a closed manifold (M, gij(t)) evolving by ∂tgij = −2Sij, where Sij is a symmetric tensor field of (2,0)-type on M. We extend some recent results of Topping, Lott and Bre ..."

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Abstract. In this note we will adapt Topping’s L-optimal transportation theory for Ricci flow to a more general situation, i.e. to a closed manifold (M, gij(t)) evolving by ∂tgij = −2Sij, where Sij 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) W-entropy, and recover the monotonicity of Müller’s (and hence also of Perelman’s) reduced volume. 1.

### PhD Project Proposal — Dr. Reto Müller Singularities of geometric flows

"... In mathematics and in particular in geometric analysis, heat flow methods have become an important and exciting tool, the motivation being to evolve rough initial data towards nice objects, e.g. manifolds with constant curvature, harmonic maps or minimal surfaces. As these geometric heat flows are s ..."

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In mathematics and in particular in geometric analysis, heat flow methods have become an important and exciting tool, the motivation being to evolve rough initial data towards nice objects, e.g. manifolds with constant curvature, harmonic maps or minimal surfaces. As these geometric heat flows are solutions of nonlinear parabolic partial differential equations, they typically develop singularities in finite or infinite time. A singularity occurs for example when the evolving object (e.g. the underlying Riemannian manifold or map) degenerates in some sense. The proposed project focuses on the study of how singularities form and on a canonical characterisation of them. In particular, the project will include work on Hamilton’s Ricci Flow, the Harmonic Map Flow, and the Harmonic Ricci Flow, a coupling of the other two introduced a few years ago in [3]. There are many possible objectives for PhD projects in this area. Here, I only mention a few of them: One goal could be to adopt and implement the state of the art knowledge on Ricci Flow singularities to the case of the Harmonic Ricci Flow. In particular, the results for fast-forming singularities (so-called Type I singularities) of the Ricci Flow [1] should translate to the coupled flow – for which the necessary monotonicity formulas have already been constructed [2]. In order to do this, the PhD student will have to extend these monotonicity formulas to a version “based at the singular time ” and then modify the proof of the main theorem in [1] to the technically more complicated situation of the coupled flow. Other, more ambitious goals include in particular trying to get a better understanding of Type II and Type III singularities of the Ricci Flow, which are much harder to analyse. Possible research directions here are the construction of explicit examples of Type II singularities which are not rotationally symmetric (all the known ones are!) and the further development of the theory of Type III Ricci Flow singularities using the Harmonic Ricci flow which naturally appears in the study of this kind of singularities.