Lagrangian Bonnet pairs in complex space forms

TL;DR

This paper establishes a Bonnet theorem for conformal Lagrangian surfaces in complex space forms and investigates the uniqueness of isometric Lagrangian surfaces with the same mean curvature form, introducing the concept of Lagrangian Bonnet pairs.

Contribution

It provides the first Bonnet theorem for conformal Lagrangian surfaces and characterizes conditions for the uniqueness of Lagrangian Bonnet pairs in complex space forms.

Findings

Any compact Lagrangian surface admits at most one other isometric Lagrangian surface with the same mean curvature form.

Lagrangian Bonnet pairs occur unless the Maslov form is conformal.

New results on Lagrangian Bonnet surfaces in $ ilde{M}^2(4c)$.

Abstract

In this paper we first give a Bonnet theorem for conformal Lagrangian surfaces in complex space forms, then we show that any compact Lagrangian surface in the complex space form admits at most one other global isometric Lagrangian surface with the same mean curvature form, unless the Maslov form is conformal. These two Lagrangian surfaces are then called Lagrangian Bonnet pairs. We also studied the question about Lagrangian Bonnet surfaces in $\tilde{M}^2(4c)$, and obtain some interesting results.