Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 41, 2016, 3-21

CONFORMAL MEASURES AND LOCALLY CONFORMALLY FLAT METRIC TENSORS

Vesa Ala-Mattila

University of Helsinki, Department of Mathematics and Statistics
P.O. Box 68, FI-00014 University of Helsinki, Finland; vesa.ala-mattila 'at' helsinki.fi

Abstract. In the paper [Na], Nayatani used a Patterson-Sullivan measure μ of a non-elementary Kleinian group Γ of the second kind to define a metric tensor g^μ on the set of discontinuity Ω(Γ) of Γ which is compatible with the natural conformal structure of Ω(Γ). The metric tensor g^μ is Γ-invariant and so it can be projected to a metric tensor g^μ_M of any Kleinian manifold M contained in the quotient Ω(Γ)/Γ. Nayatani showed that the sign of the scalar curvature of g^μ is determined by the exponent of convergence δ_Γ of Γ. He showed also that in some situations the isometry group of (M,g_M^μ) coincides with the group of conformal automorphisms of M. We point out in this paper that Nayatani's definitions and arguments can be applied if the Patterson-Sullivan measure μ is replaced by any conformal measure of Γ supported by the limit set L(Γ) of Γ. Combining this observation with an existence theorem of conformal measures proved in [AFTu] and [Sul3], we deduce that if Γ is not convex cocompact, then Γ has many metric tensors like g^μ and some of them must have scalar curvatures which are negative everywhere. We also obtain a simple new proof for the known fact that if M is compact and δ_Γ ≤ (n - 2)/2, where n ≥ 3 is the dimension of M, then Γ is convex cocompact. Finally, we point out generalizations of Nayatani's results (and results of others) regarding the isometry group of (M,g_M^μ).

2010 Mathematics Subject Classification: Primary 30F40; Secondary 53A30, 37F35.

Key words: Kleinian groups, conformal measures, Nayatani tensors, locally conformally flat Riemannian manifolds, Kleinian manifolds.

Reference to this article: V. Ala-Mattila: Conformal measures and locally conformally flat metric tensors. Ann. Acad. Sci. Fenn. Math. 41 (2016), 3-21.

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doi:10.5186/aasfm.2016.4109