Lagrangian submanifolds in para-complex Euclidean space

Abstract

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