Lagrangian submanifolds in para-complex Euclidean space

TL;DR

This paper explores the geometry of Lagrangian submanifolds in para-complex Euclidean space, focusing on curvature equations, minimality conditions, and self-similar solutions under mean curvature flow.

Contribution

It provides new insights into the extrinsic geometry of Lagrangian submanifolds in para-Kähler spaces, including classifications and examples of minimal and self-similar solutions.

Findings

Para-complex submanifolds are minimal.

Characterization of minimal Lagrangian surfaces with indefinite metric.

Description of SO(n)-equivariant Lagrangian self-similar solutions.

Abstract

We address the study of some curvature equations for distinguished submanifolds in para-K\"ahler geometry. We first observe that a para-complex submanifold of a para-K\"ahler manifold is minimal. Next we describe the extrinsic geometry of Lagrangian submanifolds in the para-complex Euclidean space D^n and discuss a number of examples, such as graphs and normal bundles. We also characterize those Lagrangian surfaces of D^2 which are minimal and have indefinite metric. Finally we describe the Lagrangian self-similar solutions of the Mean Curvature Flow which are SO(n)-equivariant.