Counting Minimal Lagrangians Via Mirzakhani Functions

Abstract

We show that for $k>1$ the number of genus $k$ minimal Lagrangians with area at most $A$ in a product of hyperbolic surfaces grows on the order of $A^{6(k-1)}$, with an explicit leading constant given in terms of the Mirzakhani function, and we obtain a similar result for products of nonpositively curved surfaces. We also prove rigidity of the Lagrangian area spectrum, and obtain analogous counting results for products of a higher genus surface with a circle.

Theorem 1.4 extended to nonpositive curvature