Systole functions and Weil–Petersson geometry

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 st...

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Bibliographic Details
Published in:Mathematische annalen 2024-06, Vol.389 (2), p.1405-1440
Main Author: Wu, Yunhui
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
Riemann surfaces
Systole
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Summary: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.
ISSN:0025-5831
1432-1807
DOI:10.1007/s00208-023-02679-7