Abstract
Starting from two Lagrangian immersions and a Legendre curve \({\tilde{\gamma}(t)}\) in \({\mathbb{S}^3(1)}\) \(({\rm or\,in}\,{\mathbb{H}_1^3(-1)})\), it is possible to construct a new Lagrangian immersion in \({\mathbb{CP}^n(4)}\) \(({\rm or\,in}\,{\mathbb{CH}^n(-4)})\), which is called a warped product Lagrangian immersion. When \({\tilde{\gamma}(t)=(r_1e^{i(\frac{r_2}{r_1}at)}, \;r_2e^{i(- \frac{r_1}{r_2}at)})}\) \(({\rm or}\,{\tilde{\gamma}(t)=(r_1e^{i(\frac{r_2}{r_1}at)}, \;r_2e^{i( \frac{r_1}{r_2}at)})})\), where r 1, r 2, and a are positive constants with \({r_1^2+r_2^2=1}\) \(({\rm or}\,{-r_1^2+r_2^2=-1})\), 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 \({\mathbb{CP}^n(4)}\) or \({\mathbb{CH}^n(-4)}\) 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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Dedicated to Heinrich Wefelscheid on the occasion of his 70th birthday
The first author was supported by NSFC grant No. 10971110 and Tsinghua University–K.U. Leuven Bilateral Scientific Cooperation Fund. The second author was supported by NSFC grant No. 10701007 and Tsinghua University–K.U. Leuven Bilateral Scientific Cooperation Fund.
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Li, H., Wang, X. Calabi Product Lagrangian Immersions in Complex Projective Space and Complex Hyperbolic Space. Results. Math. 59, 453–470 (2011). https://doi.org/10.1007/s00025-011-0107-z
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DOI: https://doi.org/10.1007/s00025-011-0107-z