Holomorphic vector fields and minimal Lagrangian submanifolds

Abstract

The purpose of this note is to establish the following theorem: Let N be a Kahler manifold, L be a compact oriented immersed minimal Lagrangian submanifold in N and V be a holomorphic vector field in a neighbourhood of L in N. Let div(V) be the (complex) divergence of V. Then the integral of div(V) over L is 0. Vice versa let N^2n be Kahler-Einstein with non-zero scalar curvature and L^n be a totally real oriented embedded n-dimensional real-analytic submanifold of N s.t. the divergence of any holomorphic vector field defined in a neighbourhood of L in N integrates to 0 on L. Then L is a minimal Lagrangian submanifold of N.