Lagrangian submanifolds of the nearly K\"ahler $\mathbb{S}^3 \times \mathbb{S}^3$ from minimal surfaces in $\mathbb{S}^3$

TL;DR

This paper explores a special class of Lagrangian submanifolds in a nearly Kähler manifold, linking their properties to minimal surfaces in the 3-sphere, and provides a method to construct many such submanifolds from minimal surfaces.

Contribution

It establishes a connection between non-totally geodesic Lagrangian submanifolds and minimal surfaces in S^3, offering a construction method for these submanifolds from minimal surfaces.

Findings

Characterization of Lagrangian submanifolds via angle functions

Construction of Lagrangian immersions from minimal surfaces

Identification of an exceptional example

Abstract

We study non-totally geodesic Lagrangian submanifolds of the nearly K\"ahler $\mathbb{S}^3 \times \mathbb{S}^3$ for which the projection on the first component is nowhere of maximal rank. We show that this property can be expressed in terms of the so called angle functions and that such Lagrangian submanifolds are closely related to minimal surfaces in $\mathbb{S}^3$. Indeed, starting from an arbitrary minimal surface, we can construct locally a large family of such Lagrangian immersions, including one exceptional example.