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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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