Lemme de l'ombre et non divergence des horosphères d'une variété géométriquement finie

Schapira, Barbara

Annales de l'Institut Fourier, Tome 54 (2004) no. 4, pp. 939-987.

Dans cet article, nous établissons dans un premier temps un lemme de l'ombre dans le cas des variétés géométriquement finies à courbure négative variable. Ce théorème donne des estimées très précises de la décroissance de la mesure de Patterson des ombres, sur le bord à l'infini de telles variétés. Nous en déduisons un résultat de non divergence des horosphères. Plus précisément, nous considérons certaines moyennes naturelles sur de grandes boules horosphériques, dont nous montrons la tension lorsque le rayon des boules tend vers l'infini; en d'autres termes, la non compacité de la variété n'entraîne pas de perte de masse pour ces moyennes.

In this work, we prove first the Shadow Lemma on geometrically finite manifolds with variable negative curvature. This result gives sharp estimates on the decreasing behavior of the Patterson measure of shadows, on the boundary at infinity of the manifold. We deduce a result of nondivergence of the horospheres of the manifold. More precisely, we prove the tightness of natural averages on large horospherical balls ; in other words, there is no loss of mass due to the lack of compacity of the manifold.

Classification : 37D40, 37C85, 28D99
Mot clés : horosphères, variété géométriquement finie, non divergence, lemme de l'ombre, mesure de Patterson
Keywords: horospheres, geometrically finite manifolds, nondivergence (or tightness), Shadow Lemma, Patterson measure

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