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Espaces de longueur d'entropie majorée : rigidité topologique, adhérence des variétés, noyau de la chaleur

Abstract : In order to obtain some precompactness and boundedness results, one generally consider the set of compact manifolds whose dimension, diameter and curvature are bounded. Since this set is not complete, there exists no general proof by compactness/ continuity. of the boundedness of invariants. As the entropy is almost unsensible to local variations of the metric or of the topology, we chose to consider the much larger family M(δ,H,D) of length spaces which admit a universal cover, whose diameter and (volume) entropy are bounded by H and D and which satisfy some 1-homotopic condition (called δ-non-abelianess). We show that M(δ,H,D) is complete w.r.t. the Gromov-Hausdorff distance; that the entropy and the marked length spectrum (resp. the first Betti number and the fondamental group) are lipschitz (resp. locally constant) functions; that the volume and the infimum of the curvature of two ε-close manifolds may be compared. Moreover, the closure of the subset M0(δ,H,D,V) (of negatively curved manifolds whose volume is bounded by V) is compact in M0(δ,H,D) and some uniform upper bounds on the heat kernel imply the precompactness of M0(δ,H,D,V) w.r.t. the spectral distance (which allows in particular a good description of the limit-spaces). The method is based on estimations of the volume of balls (without any assumptions on the curvature) and on the computation of a uniform ε := ε (δ,H,D) such that every Hausdorff ε-approximation between two spaces X and Y (which belong to M(δ,H,D) induces an isomorphism ρ between the groups of automorphisms of their universal covers, and lifts to a ρ-equivariant ε-almost-isometry between these covers.
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Submitted on : Monday, May 9, 2005 - 3:42:13 PM
Last modification on : Wednesday, November 4, 2020 - 2:10:12 PM
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Guillemette Reviron. Espaces de longueur d'entropie majorée : rigidité topologique, adhérence des variétés, noyau de la chaleur. Mathématiques [math]. Université Joseph-Fourier - Grenoble I, 2005. Français. ⟨tel-00009203⟩

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