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Théorèmes d’existence en temps court du flot de Ricci pour des variétés non-complètes, non-éffondrées, à courbure minorée.

Abstract : The Ricci Flow is a partial differential equation governing the evolution of a Riemannian metric depending on a time parameter t on a differential manifold. It was first introduced and studied by R. Hamilton, and eventually led to the solution of the Geometrization conjecture for closed three-dimensional manifolds by G. Perelman in 2001. The classical short-time existence theory for the Ricci Flow, due to Hamilton and Shi, asserts, in any dimension, the existence of a flow starting from any initial metric when the underlying manifold in compact, or for any complete initial metric with a bound on the norm of the curvature tensor otherwise. In the absence of such a bound, though, the conjecture is that starting from dimension 3 one can find such initial data for which there is no solution. In this thesis, we prove short-time existence theorems under hypotheses weaker than a bound on the norm of the curvature tensor. To do this, we introduce a general construction which, for any Riemannian metric g (not necessarily complete) on a manifold M, allows us to produce a solution to the equation of the flow on an open domain D of the space-time M * [0,T] which contains the initial time slice, with g as an initial datum. We proceed to show that under suitable hypotheses on g, one can control the shape of the domain D, so that in particular, D contains a subset of the form M * [0,t] with t>0 if g is complete. By « suitable hypothesis », we mean one of the following. In any case, we assume a lower bound on the volume of balls of radius at most 1, plus a) in dimension 3, a lower bound on the Ricci tensor, b) in dimension n, a lower bound on the so-called « isotropic curvature I » or c) in dimension n, a bound on the norm of the Ricci tensor, as well as a hypothesis which garanties the metric proximity of every ball of radius at most 1 with a ball of the same radius in a metric product between a three-dimensional metric space and a n-3 dimensional Euclidian factor. Moreover, with these existence results come estimates on the existence time and regularization properties of the flow, quantified in term of the hypotheses on the initial data. The possibility to regularize metrics, locally or globally, with such estimates has consequences in terms of the metric spaces obtained as limits, in the Gromov-Hausdorff topology, of sequences of manifolds uniformly satisfying a), b) or c). Indeed, the classical compactness theorems for the Ricci Flow allow for the extraction of a limit flow for any sequence of initial metrics uniformly satisfying the hypotheses and thus possessing a flow for a controlled amount of time. In the case when these metrics approach a singular space in the Gromov-Hausdorff topology, such a limit solution can be interpreted as a flow regularizing the singular limit space, the existence of which puts constraints on the topology of this space.
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Submitted on : Monday, April 8, 2019 - 11:53:06 AM
Last modification on : Friday, October 23, 2020 - 12:05:11 PM


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Raphaël Hochard. Théorèmes d’existence en temps court du flot de Ricci pour des variétés non-complètes, non-éffondrées, à courbure minorée.. Géométrie différentielle [math.DG]. Université de Bordeaux, 2019. Français. ⟨NNT : 2019BORD0006⟩. ⟨tel-02092609⟩



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