Some families of special Lagrangian tori

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].