Lagrangian surfaces in $\mathbb H^2 \times \mathbb H^2$

TL;DR

This paper investigates Lagrangian surfaces within the product of two hyperbolic planes, providing classifications and examples of special types such as totally geodesic, umbilical, and minimal surfaces with constant Gaussian curvature.

Contribution

It offers a comprehensive classification of various special Lagrangian surfaces in the complex hyperbolic quadric formed by the product of two hyperbolic planes.

Findings

Classified totally umbilical and totally geodesic Lagrangian surfaces

Constructed examples of Lagrangian surfaces with specific properties

Identified conditions for minimal Lagrangian surfaces with constant Gaussian curvature

Abstract

The Riemannian product of two hyperbolic planes of constant Gaussian curvature -1 has a natural K\"ahler structure. In fact, it can be identified with the complex hyperbolic quadric of complex dimension two. In this paper we study Lagrangian surfaces in this manifold. We present several examples and classify the totally umbilical and totally geodesic Lagrangian surfaces, the Lagrangian surfaces with parallel second fundamental form, the minimal Lagrangian surfaces with constant Gaussian curvature and the complete minimal Lagrangian surfaces satisfying a bounding condition on an important function that can be defined on any Lagrangian surface in this particular ambient space.