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Canonical metric on the domain of discontinuity of a Kleinian group. (English) Zbl 0979.53036
Séminaire de théorie spectrale et géométrie. Année 1997-1998. St. Martin D’Hères: Université de Grenoble I, Institut Fourier, Sémin. Théor. Spectrale Géom., Chambéry-Grenoble. 16, 9-32 (1998).
From the introduction: A Kleinian group is a discrete subgroup of the conformal automorphism group of the round sphere. Its domain of discontinuity is by definition the largest open subset of the sphere on which the group acts properly discontinuously. The quotient of the domain by the group inherits the flat conformal structure of the sphere. In [Math. Z. 225, 115-131 (1997; Zbl 0868.53024)] the second author introduced a canonical Riemannian metric on such a manifold which is compatible with the conformal structure. He observed that the curvature of this metric well reflects the Hausdorff dimension of the limit set of the Kleinian group. This recovers R. Schoen and S.-T. Yau’s earlier result [Invent. Math. 92, 47-71 (1988; Zbl 0658.53038)] on the relation between the Yamabe conformal invariant of the quotient manifold and the Hausdorff dimension of the limit set. Our result roughly states that the smaller the dimension of the limit set, the stronger the positivity of curvature. Via the classical Bochner technique, this leads to a vanishing theorem for the cohomology of the quotient manifold. The first author [Invent. Math. 122, 603-625 (1995; Zbl 0854.53035)] then used this vanishing result to generalize R. Bowen’s theorem [Publ. Math., Inst. Hautes Etud. Sci. 50, 11-25 (1979; Zbl 0439.30032)] on the Hausdorff dimension of the limit set of a quasi-Fuchsian group to higher dimensions.
This article surveys various aspects of the canonical metric, and as such it is partly expository. It, however, also contains new results which we have obtained after the writing of [the first author, loc. cit.], [the second author, loc. cit. and Rep. First MSJ Int. Res. Inst., Tohoku Univ. Math. Inst. 341-349 (1993; Zbl 1040.53502)].
For the entire collection see [Zbl 0904.00015].

MSC:
53C20 Global Riemannian geometry, including pinching
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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