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- Title
Systole functions and Weil–Petersson geometry.
- Authors
Wu, Yunhui
- Abstract
A basic feature of Teichmüller theory of Riemann surfaces is the interplay of two dimensional hyperbolic geometry, the behavior of geodesic-length functions and Weil–Petersson geometry. Let T g (g ⩾ 2) be the Teichmüller space of closed Riemann surfaces of genus g. Our goal in this paper is to study the gradients of geodesic-length functions along systolic curves. We show that their L p (1 ⩽ p ⩽ ∞) -norms at every hyperbolic surface X ∈ T g are uniformly comparable to ℓ sys (X) 1 p where ℓ sys (X) is the systole of X. As an application, we show that the minimal Weil–Petersson holomorphic sectional curvature at every hyperbolic surface X ∈ T g is bounded above by a uniform negative constant independent of g, which negatively answers a question of Mirzakhani. Some other applications to the geometry of T g will also be discussed.
- Subjects
TEICHMULLER spaces; HYPERBOLIC geometry; RIEMANN surfaces; GEOMETRY; CURVATURE
- Publication
Mathematische Annalen, 2024, Vol 389, Issue 2, p1405
- ISSN
0025-5831
- Publication type
Article
- DOI
10.1007/s00208-023-02679-7