Localization of weighted Goldman volume on SL(3,R)-Hitchin Component
Abstract
We compute Goldman volume on the moduli space of convex real projective structures on a closed oriented surface of genus at least two, with exponential weight defined by the shifted Blaschke area. For every positive parameter, the integral converges and is a finite sum of inverse powers of that parameter. Its coefficients are intersection numbers involving the bundle of regular cubic differentials on the moduli stack of stable curves. The proof establishes absolute convergence and vanishing signed boundary integrals of the transgression under simultaneous plumbing degeneration. The pushforward of Goldman measure by the shifted Blaschke area therefore has a polynomial density on [0,∞), and its sublevel volumes are polynomial in the threshold.
Related
- The Thick Part of the PSLn(ℝ)-Hitchin-Riemann Moduli Space has Infinite Volume
- Volume of area-bounded PGL3-Hitchin moduli space
- On the Asymptotics of the Volume of Hitchin Moduli Spaces
- Localization and weighted theory on the Bergman spaces
- Component-weighted sparse group LASSO for finite mixture regression models
BibTeX
@misc{cao2026localization,
title = {{Localization of weighted Goldman volume on $\mathrmSL(3,\R)$-Hitchin Component}},
author = {Shiyu Cao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37317},
eprint = {2610.37317},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37317},
}
RIS
TY - UNPB
TI - Localization of weighted Goldman volume on SL(3,R)-Hitchin Component
AU - Shiyu Cao
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37317
UR - https://arxiv.org/abs/2610.37317
AB - We compute Goldman volume on the moduli space of convex real projective structures on a closed oriented surface of genus at least two, with exponential weight defined by the shifted Blaschke area. For every positive parameter, the integral converges and is a finite sum of inverse powers of that parameter. Its coefficients are intersection numbers involving the bundle of regular cubic differentials on the moduli stack of stable curves. The proof establishes absolute convergence and vanishing signed boundary integrals of the transgression under simultaneous plumbing degeneration. The pushforward of Goldman measure by the shifted Blaschke area therefore has a polynomial density on [0,∞), and its sublevel volumes are polynomial in the threshold.
ER -