Uniform asymptotics and the Weil--Petersson constant
Abstract
In their celebrated work on the large-genus asymptotic expansion of Weil--Petersson volumes, Mirzakhani and Zograf left open the question of whether the Weil--Petersson constant satisfies CWP=π−1/2. We resolve the conjecture affirmatively. Our strategy is to consider a family of spectral curves interpolating between y=z−4z3 and y=sin(2πz)/(2π). For each fixed number n≥2 of marked points, we normalize the cusp coefficients using the action integral and the derivative at its first positive zero. We prove that the logarithmic derivatives of these normalized coefficients with respect to the family parameter are On(g−1), uniformly along the family. The proof combines the descendant recursion, positivity, the dilaton equation, KdV, and coefficient asymptotics for the Painlevé I equation.
Related
- Uniform large-genus asymptotics of Weil-Petersson volumes
- Generalized Hitchin-Kobayashi correspondence and Weil-Petersson current
- Closed Response Calculus for SLE Weldings and Weil--Petersson Kähler Geometry
- Scalar curvature, volume, and Weil-Petersson translation length
- The Weil-Petersson diameter of moduli space for large genus
BibTeX
@misc{liu2026uniform,
title = {{Uniform asymptotics and the Weil--Petersson constant}},
author = {Kefeng Liu and Hao Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33752},
eprint = {2610.33752},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33752},
}
RIS
TY - UNPB
TI - Uniform asymptotics and the Weil--Petersson constant
AU - Kefeng Liu
AU - Hao Xu
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33752
UR - https://arxiv.org/abs/2610.33752
AB - In their celebrated work on the large-genus asymptotic expansion of Weil--Petersson volumes, Mirzakhani and Zograf left open the question of whether the Weil--Petersson constant satisfies C_WP=π^−1/2. We resolve the conjecture affirmatively. Our strategy is to consider a family of spectral curves interpolating between y=z−4z^3 and y=sin(2πz)/(2π). For each fixed number n≥2 of marked points, we normalize the cusp coefficients using the action integral and the derivative at its first positive zero. We prove that the logarithmic derivatives of these normalized coefficients with respect to the family parameter are O_n(g^−1), uniformly along the family. The proof combines the descendant recursion, positivity, the dilaton equation, KdV, and coefficient asymptotics for the Painlevé I equation.
ER -