On the existence of geodesic connecting Lagrangian graphs in ${\mathbb{C}}^n$

In this paper we show that two Lagrangian graphs over the torus in ${\mathbb{C}}^n$ with large Lagrangian phase can be connected via Lipschitz continuous geodesic with respect to the $L^2$ metric on the space of Lagrangian submanifolds. In particular, the geodesic for Lagrangian graphs over the torus in ${\mathbb{C}}^n$ can be formulated as a degenerate elliptic equation, and we construct geodesic by solving the corresponding Dirichlet problem.