Finiteness of algebraic Weil-Petersson totally geodesic curves in pointed moduli spaces
Abstract
For every pair (g,n) with 2g−2+n>0, we prove that ℳg,n contains only finitely many closed algebraic curves that are totally geodesic for the Weil-Petersson metric. We show that every such curve is an arithmetic quadratic Teichmüller curve and has equal heights for all marked maximal cylinders in every periodic direction. Although there are infinitely many arithmetic Teichmüller curves, we can bound the degree of the associated torus covering by the crucial equal-height property. This leads to the desired finiteness statement. We also prove that quadratic Teichmüller curves are characterized by the existence of a holomorphic splitting of the logarithmic cotangent Kodaira-Spencer map. This result is the analogue of Möller and Viehweg's characterization of Kobayashi geodesics in 𝒜g.
Related
- Quantitative finiteness for maximal complex totally geodesic submanifolds of non-arithmetic ball quotients
- Totally Geodesic Submanifolds of Teichmüller Space in Complex Dimension at Least Three
- Rigidity of Constant-Scalar-Curvature Ricci Shrinkers with Totally Geodesic Focal Set
- Arithmeticity, superrigidity, and totally geodesic submanifolds of infinite-volume hyperbolic manifolds
- Finiteness theorems for trace Euclidean quadratic lattices over totally real number fields
BibTeX
@misc{hu2026finiteness,
title = {{Finiteness of algebraic Weil-Petersson totally geodesic curves in pointed moduli spaces}},
author = {Tianzhi Hu and Kang Zuo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35086},
eprint = {2610.35086},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35086},
}
RIS
TY - UNPB
TI - Finiteness of algebraic Weil-Petersson totally geodesic curves in pointed moduli spaces
AU - Tianzhi Hu
AU - Kang Zuo
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35086
UR - https://arxiv.org/abs/2610.35086
AB - For every pair (g,n) with 2g−2+n>0, we prove that ℳ_g,n contains only finitely many closed algebraic curves that are totally geodesic for the Weil-Petersson metric. We show that every such curve is an arithmetic quadratic Teichmüller curve and has equal heights for all marked maximal cylinders in every periodic direction. Although there are infinitely many arithmetic Teichmüller curves, we can bound the degree of the associated torus covering by the crucial equal-height property. This leads to the desired finiteness statement. We also prove that quadratic Teichmüller curves are characterized by the existence of a holomorphic splitting of the logarithmic cotangent Kodaira-Spencer map. This result is the analogue of Möller and Viehweg's characterization of Kobayashi geodesics in 𝒜_g.
ER -