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Search: MSC category 53D12 ( Lagrangian submanifolds; Maslov index )

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1. CMB Online first

Gupta, Purvi
A real-analytic nonpolynomially convex isotropic torus with no attached discs
We show by means of an example in $\mathbb C^3$ that Gromov's theorem on the presence of attached holomorphic discs for compact Lagrangian manifolds is not true in the subcritical real-analytic case, even in the absence of an obvious obstruction, i.e, polynomial convexity.

Keywords:polynomial hull, isotropic submanifold, holomorphic disc
Categories:32V40, 32E20, 53D12

2. CMB 2007 (vol 50 pp. 321)

Blair, David E.
On Lagrangian Catenoids
Recently I. Castro and F. Urbano introduced the Lagrangian catenoid. Topologically, it is $\mathbb R\times S^{n-1}$ and its induced metric is conformally flat, but not cylindrical. Their result is that if a Lagrangian minimal submanifold in ${\mathbb C}^n$ is foliated by round $(n-1)$-spheres, it is congruent to a Lagrangian catenoid. Here we study the question of conformally flat, minimal, Lagrangian submanifolds in ${\mathbb C}^n$. The general problem is formidable, but we first show that such a submanifold resembles a Lagrangian catenoid in that its Schouten tensor has an eigenvalue of multiplicity one. Then, restricting to the case of at most two eigenvalues, we show that the submanifold is either flat and totally geodesic or is homothetic to (a piece of) the Lagrangian catenoid.

Categories:53C42, 53D12

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