Du
Shedule -
Équations différentielles motiviques et au–delà
Exponential volumes of moduli spaces of hyperbolic surfaces and recursions
IHP, Paris
salle 314
This is a joint work with Zhe Sun.
A decorated surface S is an oriented topological surface with boundary, equipped with marked points on the boundary considered modulo the isotopy.
We consider the moduli space M(S) of hyperbolic structures on S with geodesic boundary, such that the hyperbolic structure near each marked point is a cusp, equipped with a horocycle. The space M(S) carries a canonical volume form. However, if the cusps are present, the volume of the space M(S), as well as its variant without horocycles, are infinite.
We introduce the exponential volume form, given by the volume form multiplied by the exponent of a canonical function on M(S).
We show that the exponential volume is finite. We prove recursion formulas for the exponential volumes, generalising Mirzakhani's recursions for the volumes of moduli spaces of hyperbolic surfaces.
We consider the moduli space M(S) of hyperbolic structures on S with geodesic boundary, such that the hyperbolic structure near each marked point is a cusp, equipped with a horocycle. The space M(S) carries a canonical volume form. However, if the cusps are present, the volume of the space M(S), as well as its variant without horocycles, are infinite.
We introduce the exponential volume form, given by the volume form multiplied by the exponent of a canonical function on M(S).
We show that the exponential volume is finite. We prove recursion formulas for the exponential volumes, generalising Mirzakhani's recursions for the volumes of moduli spaces of hyperbolic surfaces.
We suggest that the moduli space M(S) with the exponential volume form is the true analog of the moduli space M_{g,n}, relevant to the open string theory.