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On a minimal Lagrangian submanifold of \(\mathbf C^n\) foliated by spheres. (English) Zbl 0974.53059
The Lagrangian catenoid in \(\mathbb{C}^n\) is defined by \(M_0=\{(x,y)\in\mathbb{C}^n=\mathbb{R}^n \times \mathbb{R}^n : |x|y=|y|x\), \(\text{Im}(|x|+i|y|)^n=1\), \(|y|<|x|\tan(\pi/n)\}\). The authors prove the following interesting characterizations of the Lagrangian catenoid.
Theorem 1. Let \(\phi: M\to \mathbb{C}^n\) be a minimal (nonflat) Lagrangian immersion of an \(n\)-manifold. Then \(M\) is foliated by pieces of round \((n-1)\)-spheres of \(\mathbb{C}^n\) if and only if \(\phi\) is congruent (up to dilations) to an open subset of the Lagrangian catenoid.
Theorem 2. Let \(\phi: M\to \mathbb{C}^m\) be a (nonflat) complex immersion of a complex \(n\)-dimensional Kähler manifold \(M\). Then \(M\) is foliated by pieces of round \((2n-1)\)-spheres of \(\mathbb{C}^m\) if and only if \(n=1\) and \(\phi\) is congruent (up to dilations) to an open subset of the Lagrangian catenoid.
Theorem 3. Let \(M\) be an \(n\)-dimensional \((n\geq 3)\) complete minimal (nonflat) submanifold with finite total scalar curvature immersed in Euclidean space \({\mathbf E}^{2n}\). Then the compactification by the inversion of \(M\) is Lagrangian for a certain orthogonal complex structure on \({\mathbf E}^{2n}\) if and only if \(M\) is (up to dilations) the Lagrangian catenoid.

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