Calabi product Lagrangian immersions in complex projective space and complex hyperbolic space

TL;DR

This paper investigates how to identify Calabi product Lagrangian immersions in complex projective and hyperbolic spaces based on second fundamental form properties, providing characterizations for minimal, Hamiltonian minimal, and parallel cases.

Contribution

It offers criteria to determine when a Lagrangian immersion is a Calabi product, especially in special cases like minimality and parallel second fundamental form.

Findings

Characterization of Calabi product Lagrangian immersions based on second fundamental form.

Criteria for minimal, Hamiltonian minimal, and parallel second fundamental form cases.

Methods to identify Calabi product structure from geometric properties.

Abstract

Starting from two Lagrangian immersions and a Legendre curve $\tilde{\gamma}(t)$ in $\mathbb{S}^3(1)$ (or in $\mathbb{H}_1^3(1)$), it is possible to construct a new Lagrangian immersion in $\mathbb{CP}^n$ (or in $\mathbb{CH}^n$), which is called a warped product Lagrangian immersion. When $\tilde{\gamma}(t)=(r_1e^{i(\frac{r_2}{r_1}at)}, r_2e^{i(- \frac{r_1}{r_2}at)})$ (or $\tilde{\gamma}(t)=(r_1e^{i(\frac{r_2}{r_1}at)}, r_2e^{i(\frac{r_1}{r_2}at)})$), where $r_1$, $r_2$, and $a$ are positive constants with $r_1^2+r_2^2=1$ (or $-r_1^2+r_2^2=-1$), we call the new Lagrangian immersion a Calabi product Lagrangian immersion. In this paper, we study the inverse problem: how to determine from the properties of the second fundamental form whether a given Lagrangian immersion of $\mathbb{CP}^n$ or $\mathbb{CH}^n$ is a Calabi product Lagrangian immersion. When the Calabi product is minimal, or is Hamiltonian minimal, or has parallel second fundamental form, we give some further characterizations.