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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.)
##### Keywords:
Lagrangian submanifold; Lagrangian catenoid
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