The number of genus-k minimal Lagrangians of area ≤ A in a product of hyperbolic surfaces grows asymptotically as c A^{6(k-1)} where c is an explicit constant given in terms of the Mirzakhani function.
Thurston,Minimal stretch maps between hyperbolic surfaces, arXiv:math/9801039
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abstract
This paper develops a theory of Lipschitz comparisons of hyperbolic surfaces analogous to the theory of quasi-conformal comparisons. Extremal Lipschitz maps (minimal stretch maps) and geodesics for the `Lipschitz metric' are constructed. The extremal Lipschitz constant equals the maximum ratio of lengths of measured laminations, which is attained with probability one on a simple closed curve. Cataclysms are introduced, generalizing earthquakes by permitting more violent shearing in both directions along a fault. Cataclysms provide useful coordinates for Teichmuller space that are convenient for computing derivatives of geometric function in Teichmuller space and measured lamination space.
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Characterizes ∞-harmonic maps (critical points of L^∞ derivative norm) via 1-currents on the domain that are critical for a generalised mass functional.
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Counting Minimal Lagrangians Via Mirzakhani Functions
The number of genus-k minimal Lagrangians of area ≤ A in a product of hyperbolic surfaces grows asymptotically as c A^{6(k-1)} where c is an explicit constant given in terms of the Mirzakhani function.
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A characterisation of $\infty$-harmonic maps in terms of $1$-currents
Characterizes ∞-harmonic maps (critical points of L^∞ derivative norm) via 1-currents on the domain that are critical for a generalised mass functional.