Let D be a smoothly bounded pseudoconvex domain in C n , n > 1. Using the Robin function Λ(p) that arises from the Green function G(z, p) for D with pole at p ∈ D associated with the standard sum-of-squares Laplacian, N. Levenberg and H. Yamaguchi had constructed a Kähler metric (the so-called Λ-metric) on D. Assume that D is strongly pseudoconvex and ds 2 denotes the Λ-metric on D. In this article, first we prove that the holomorphic sectional curvature of ds 2 along normal directions converges to a negative constant near the boundary of D. Then, we prove that if D is not simply connected, then any nontrivial homotopy class of π 1 (D) contains a closed geodesic for ds 2 . Finally, we prove that the diminesion of the space of square integrable harmonic (p, q)-forms on D relative to ds 2 is zero except when p + q = n in which case it is infinite.
Let D be a smoothly bounded pseudoconvex domain in C n , n > 1. Using the Robin function Λ(p) that arises from the Green function G(z, p) for D with pole at p ∈ D associated with the standard sum-of-squares Laplacian, N. Levenberg and H. Yamaguchi had constructed a Kähler metric (the so-called Λ-metric) on D. In this article, we study the existence of geodesic spirals for this metric.
We compute an asymptotic expression for the Arakelov self-intersection number of the relative dualizing sheaf of Edixhoven's minimal regular model for the modular curve X 0 (p 2 ) over Q. The computation of the self-intersection numbers are used to prove an effective version of the Bogolomov conjecture for the semi-stable models of modular curves X 0 (p 2 ) and obtain a bound on the stable Faltings height for those curves in a companion article [6].
For a domain D ⊂ C n , n ≥ 2, let F k D (z) = KD(z)λ I k D (z) , where KD(z) is the Bergman kernel of D along the diagonal and λ I k D (z) is the Lebesgue measure of the Kobayashi indicatrix at the point z. This biholomorphic invariant was introduced by B locki and in this note, we study the boundary behaviour of F k D (z) near a finite type boundary point where the boundary is smooth, pseudoconvex with the corank of its Levi form being at most 1. We also compute its limiting behaviour near the boundary of certain other basic classes of domains.1991 Mathematics Subject Classification. 32F45, 32A07, 32A25.
Abstract. We study several quantities associated to the Green's function of a multiply connected domain in the complex plane. Among them are some intrinsic properties such as geodesics, curvature, and L 2 -cohomology of the capacity metric and critical points of the Green's function. The principal idea used is an affine scaling of the domain that furnishes quantitative boundary behaviour of the Green's function and related objects.
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