Volume of area-bounded PGL3-Hitchin moduli space

Abstract

For the ${\mathrm{PGL}}_3$-Hitchin moduli space of unmarked convex ${\mathbb{RP}}^2$ structures on the surface $S_{g,m}$ with fixed boundary data $\mathbf{L}$ when $m>0$, we investigate the subset whose canonical Hilbert--Busemann area is bounded above by $t$. We show that the Goldman symplectic volume of this subset is at most $C_{g,m}(t+{\lambda }^2{)}^{8g-8+3m}$, where $\lambda$ is the largest total-root boundary length for $m>0$ and is set to $0$ for $m=0$. As a consequence, We obtain the explicit upper bound for the $e^{-s\cdot \mathrm{Area}}$-weighted volume.

arXiv comment25 pages, 7 figures. This preprint supersedes the earlier preprint arXiv:2001.01295