Geodesic
On Lagrangian Catenoids
Canadian mathematical bulletin, Tome 50 (2007) no. 3, pp. 321-333

Voir la notice de l'article provenant de la source Cambridge University Press

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 $\left( n-1 \right)$ -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.
DOI : 10.4153/CMB-2007-031-4
Mots-clés : 53C42, 53D12 
Blair, David E. On Lagrangian Catenoids. Canadian mathematical bulletin, Tome 50 (2007) no. 3, pp. 321-333. doi: 10.4153/CMB-2007-031-4
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