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

TL;DR

This paper demonstrates the existence of Lipschitz continuous geodesics connecting certain Lagrangian graphs in complex Euclidean space, formulated through a degenerate elliptic equation and constructed via solving a Dirichlet problem.

Contribution

It establishes the existence of geodesics between Lagrangian graphs with large phase in ^n, using a novel approach involving degenerate elliptic equations.

Findings

Existence of Lipschitz geodesics between Lagrangian graphs

Formulation of geodesics as degenerate elliptic equations

Construction of geodesics via solving Dirichlet problems

Abstract

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.