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Lagrangian submanifolds of the complex hyperbolic quadric
Annali di Matematica Pura ed Applicata ( IF 1 ) Pub Date : 2021-01-08 , DOI: 10.1007/s10231-020-01063-5
Joeri Van der Veken , Anne Wijffels

We consider the complex hyperbolic quadric \({Q^*}^n\) as a complex hypersurface of complex anti-de Sitter space. Shape operators of this submanifold give rise to a family of local almost product structures on \({Q^*}^n\), which are then used to define local angle functions on any Lagrangian submanifold of \({Q^*}^n\). We prove that a Lagrangian immersion into \({Q^*}^n\) can be seen as the Gauss map of a spacelike hypersurface of (real) anti-de Sitter space and relate the angle functions to the principal curvatures of this hypersurface. We also give a formula relating the mean curvature of the Lagrangian immersion to these principal curvatures. The theorems are illustrated with several examples of spacelike hypersurfaces of anti-de Sitter space and their Gauss maps. Finally, we classify some families of minimal Lagrangian submanifolds of \({Q^*}^n\): those with parallel second fundamental form and those for which the induced sectional curvature is constant. In both cases, the Lagrangian submanifold is forced to be totally geodesic.



中文翻译:

复双曲二次曲面的拉格朗日子流形

我们将复双曲二次曲面\({Q ^ *} ^ n \)视为复反de Sitter空间的复超曲面。此子流形的形状算子在\({Q ^ *} ^ n \)上产生一族局部几乎乘积结构,然后用于定义\({Q ^ *} ^的任何拉格朗日子流形上的局部角函数n \)。我们证明拉格朗日浸入\({Q ^ *} ^ n \)可以看作是(实)反de Sitter空间的类空超曲面的高斯图,并将角度函数与该超曲面的主曲率相关。我们还给出了一个公式,将拉格朗日浸没的平均曲率与这些主曲率联系起来。用反de Sitter空间的类空间超曲面及其高斯图的几个示例说明了这些定理。最后,我们对\({Q ^ *} ^ n \)的最小拉格朗日子流形的一些族进行分类:那些具有平行第二基本形式,并且其诱导截面曲率是恒定的。在这两种情况下,拉格朗日子流形都被迫完全测地线。

更新日期:2021-01-10
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