Some families of special Lagrangian tori |
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Authors: | Diego Matessi |
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Institution: | (1) Centre de Recherches Mathématiques, Université de Montréal, Pavillon André-Aisenstadt, 2920 Chemin de la tour (bureau 5357), Montréal, Québec H3T 1J8, Canada (e-mail: matessi@crm.umontreal.ca), CA |
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Abstract: | We give an explicit proof of the local version of Bryant's result 1], stating that any 3-dimensional real-analytic Riemannian
manifold can be isometrically embedded as a special Lagrangian submanifold in a Calabi-Yau manifold. We then refine the theorem
proving that a certain class of real-analytic one-parameter families of metrics on a 3-torus can be isometrically embedded
in a Calabi-Yau manifold as a one-parameter family of special Lagrangian submanifolds. Two applications of these results show
how the geometry of the moduli space of 3-dimesional special Lagrangian submanifolds differs considerably from the 2-dimensional
one. First of all, applying Bryant's theorem and a construction due to Calabi we show that nearby elements of the local moduli
space of a special Lagrangian 3-torus can intersect themselves. Secondly, we use our examples of one-parameter families to
show that in dimension three (and higher) the moduli space of special Lagrangian tori is not, in general, special Lagrangian
in the sense of Hitchin 13].
Received: 18 December 2001 / Revised version: 31 January 2002 / Published online: 16 October 2002
Mathematics Subject Classification (2000): 53-XX, 53C38 |
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Keywords: | |
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