Abstract
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.
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The first author is supported by the Excellence of Science Project G0H4518N of the Belgian government, and both authors are supported by Project 3E160361 of the KU Leuven Research Fund.
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Van der Veken, J., Wijffels, A. Lagrangian submanifolds of the complex hyperbolic quadric. Annali di Matematica 200, 1871–1891 (2021). https://doi.org/10.1007/s10231-020-01063-5
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DOI: https://doi.org/10.1007/s10231-020-01063-5