On a new construction of special Lagrangian immersions in complex Euclidean space

Abstract

Special Lagrangian submanifolds of complex Euclidean space C have been studied widely over the last few years. These submanifolds are volume minimizing and, in particular, they are minimal submanifolds. When n = 2, special Lagrangian surfaces of C are exactly complex surfaces with respect to another orthogonal complex structure on R ≡ C. A very important problem here, and even a good starting point to study the not well understood problem about singularities of special Lagrangian n-folds in Calabi-Yau n-folds, is finding (non-trivial) examples of special Lagrangian submanifolds in C, since those are locally modeled on singularities of these. In [2], R. Harvey and H.B. Lawson constructed the first examples in C, where we point out the Lagrangian catenoid ([1, Remark 1], [3, Theorem A] and [5, Theorem 6.4]) to show a method of construction of special Lagrangian submanifolds of C using an (n-1)-dimensional oriented minimal Legendrian submanifold of S2n−1 and certain plane curves (for a better understanding, see Proposition A in section 2). Making use of this method, we include in section 2 three new examples of special Lagrangian submanifolds which have not been exposed yet. In [5], more examples of special Lagrangian submanifolds of C were constructed, all of them with cohomogeneity one, that is, the orbits of the symmetry group of the submanifold are of codimension one. Examples of special Lagrangian cones of cohomogeneity two were given in [3] and [5]. ∗Research partially supported by a MCYT and FEDER grant BFM2001-2967