Complex analytic properties of minimal Lagrangian submanifolds

TL;DR

This paper investigates the complex geometric properties of minimal Lagrangian submanifolds in Kähler manifolds, revealing that in negatively curved spaces, such submanifolds cannot be filled with holomorphic discs, using advanced holomorphic curve methods.

Contribution

It establishes a new curvature-dependent property of minimal Lagrangian submanifolds, showing the non-existence of holomorphic disc fillings in negative curvature settings.

Findings

Minimal Lagrangians in negative curvature spaces do not admit holomorphic disc fillings.

The proof combines holomorphic curve techniques with convexity results.

The study links ambient curvature to complex geometric properties of submanifolds.

Abstract

In this article we study complex properties of minimal Lagrangian submanifolds in Kaehler ambient spaces, and how they depend on the ambient curvature. In particular, we prove that, in the negative curvature case, minimal Lagrangians do not admit fillings by holomorphic discs. The proof relies on a mix of holomorphic curve techniques and on certain convexity results.