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Minimal Lagrangian submanifolds in the complex hyperbolic space. (English) Zbl 1032.53052
Subtle examples of minimal Lagrangian submanifolds of the complex hyperbolic space \(\mathbb{C}\mathbb{H}^n\) with large symmetry groups are exhibited. In particular, the isometry groups of \(S^{n-1}\), \(\mathbb{R}\mathbb{H}^{n-1}\) and \(\mathbb{R}^{n-1}\) acting on \(\mathbb{C}\mathbb{H}^n\) as holomorphic isometries are considered in detail. These submanifolds are characterized as the only minimal submanifolds in \(\mathbb{C}\mathbb{H}^n\) that are foliated by umbilical hypersurfaces of Lagrangian subspace \(\mathbb{R}\mathbb{H}^n\) of \(\mathbb{C}\mathbb{H}^n\). Also, families in \(\mathbb{C}\mathbb{H}^n\) of such minimal submanifolds are constructed from curves in \(\mathbb{C}\mathbb{H}^1\) and \((n-1)\)-dimensional minimal Lagrangian submanifolds of complex space forms \(\mathbb{C}\mathbb{P}^{n-1}\), \(\mathbb{C}\mathbb{H}^{n-1}\) and \(\mathbb{C}^{n-1}\). It is noted that similar results can be obtained for the complex projective space \(\mathbb{C}\mathbb{P}^n\), though there are naturally fewer families of similar submanifolds.
Reviewer: T.Okubo (Victoria)

MSC:
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53D12 Lagrangian submanifolds; Maslov index
53B25 Local submanifolds
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