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

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

2010-10-05

arxiv.org/abs/1010.0956v1

Mathematics

Differential Geometry

Scientific paper

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.

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