The asymptotic Weil-Petersson form and intersection theory on M_{g,n}

Abstract

Moduli spaces of hyperbolic surfaces with geodesic boundary components of fixed lengths may be endowed with a symplectic structure via the Weil-Petersson form. We show that, as the boundary lengths are sent to infinity, the Weil-Petersson form converges to a piecewise linear form first defined by Kontsevich. The proof rests on the observation that a hyperbolic surface with large boundary lengths resembles a graph after appropriately scaling the hyperbolic metric. We also include some applications to intersection theory on moduli spaces of curves.

22 pages

References in corpus (1)

• Convexity of geodesic-length functions: a reprise

Cited by in corpus (2)

• Moduli spaces of hyperbolic surfaces and their Weil-Petersson volumes
• Local topological recursion governs the enumeration of lattice points in ${\overline{\mathcal{M}}}_{g,n}$