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# Mathematical Research Letters

## Volume 20 (2013)

### Number 3

### Holomorphic cubic differentials and minimal Lagrangian surfaces in $\mathbb{CH}^2$

Pages: 501 – 520

DOI: http://dx.doi.org/10.4310/MRL.2013.v20.n3.a8

#### Authors

#### Abstract

Minimal Lagrangian submanifolds of a Kähler manifold represent a very interesting class of submanifolds as they are Lagrangian with respect to the symplectic structure of the ambient space, while minimal with respect to the Riemannian structure. In this paper, we study minimal Lagrangian immersions of the universal cover of closed surfaces (of genus $g \geq 2)$ in $\mathbb{CH}^2$, with prescribed data $(\sigma, tq)$, where $\sigma$ is a conformal structure on the surface $S$, and ${qdz}^3$ is a holomorphic cubic differential on the Riemann surface $(S, \sigma)$. We show existence and non-uniqueness of such minimal Lagrangian immersions. We analyze the asymptotic behaviors for such immersions, and establish the surface area with respect to the induced metric as a Weil-Petersson potential function for the space of holomorphic cubic differentials on $(S, \sigma)$.

#### 2010 Mathematics Subject Classification

Primary 53C42. Secondary 35J61, 53D12.