Counting Minimal Lagrangians Via Mirzakhani Functions

TL;DR

This paper establishes asymptotic counts for minimal Lagrangians in hyperbolic surface products, revealing growth rates and rigidity properties linked to Mirzakhani functions.

Contribution

It provides explicit asymptotic formulas for counting minimal Lagrangians and demonstrates rigidity of the Lagrangian area spectrum.

Findings

Number of genus k minimal Lagrangians grows as A^{6(k-1)}.

Explicit leading constant involves Mirzakhani function.

Proves rigidity of the Lagrangian area spectrum.

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. We also prove rigidity of the Lagrangian area spectrum, and obtain analogous counting results for products of a higher genus surface with a circle.