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On Lagrangian Catenoids

  Published:2007-09-01
 Printed: Sep 2007
  • David E. Blair
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Abstract

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.
MSC Classifications: 53C42, 53D12 show english descriptions Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]
Lagrangian submanifolds; Maslov index
53C42 - Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]
53D12 - Lagrangian submanifolds; Maslov index
 

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