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- Title
Asymptotics of Kähler–Einstein metrics on complex hyperbolic cusps.
- Authors
Fu, Xin; Hein, Hans-Joachim; Jiang, Xumin
- Abstract
Let L be a negative holomorphic line bundle over an (n - 1) -dimensional complex torus D. Let h be a Hermitian metric on L such that the curvature form of the dual Hermitian metric defines a flat Kähler metric on D. Then h is unique up to scaling, and, for some closed tubular neighborhood V of the zero section D ⊂ L , the form ω h = - (n + 1) i ∂ ∂ ¯ log (- log h) defines a complete Kähler–Einstein metric on V \ D with Ric (ω h) = - ω h . In fact, ω h is complex hyperbolic, i.e., the holomorphic sectional curvature of ω h is constant, and ω h has the usual doubly-warped cusp structure familiar from complex hyperbolic geometry. In this paper, we prove that if U is another closed tubular neighborhood of the zero section and if ω is a complete Kähler–Einstein metric with Ric (ω) = - ω on U \ D , then there exist a Hermitian metric h as above and a δ ∈ R + such that ω - ω h = O (e - δ - log h ) to all orders with respect to ω h as h → 0 . This rate is doubly exponential in the distance from a fixed point, and is sharp.
- Subjects
EINSTEIN, Albert, 1879-1955; HYPERBOLIC geometry; HERMITIAN forms; CUSP forms (Mathematics); EINSTEIN manifolds; TORUS; CURVATURE; NEIGHBORHOODS
- Publication
Calculus of Variations & Partial Differential Equations, 2024, Vol 63, Issue 1, p1
- ISSN
0944-2669
- Publication type
Article
- DOI
10.1007/s00526-023-02613-4