Special Lagrangian cones.

*(English)*Zbl 1074.53067The author studies the simplest isolated singularities of special Lagrangian varieties - cones in \(\mathbb{C}^n\) with an isolated singularity. These are also local models for more general singularities in that they are possible tangent cones to special Lagrangian currents at singular points. After recalling basic facts about special Lagrangian geometry in \(\mathbb{C}^n\), one introduces the notion of special Legendrian in \(S^{2n-1}\) and characterizes special Lagrangian cones in terms of special Legendrian links. Next, one studies the relation of the harmonic maps with minimal surfaces and the appearance of the C. Neumann system in \(S^1\)-equivariant harmonic maps into spheres. One asks which solutions of the Neumann system give rise to special Legendrian immersions, one gives explicit parametrizations of these solutions, one studies the geometry of these immersions and one obtains the periodicity conditions for the same immersions.

By using these concepts and results, an infinite family of special Lagrangians in \(\mathbb{C}^3\) is constructed, each of which has a toroidal link, obtaining a detailed geometric description of these tori. Then one constructs a one-parameter family of asymptotically conical special submanifolds from any special Lagrangian cones. One discusses some more general singularities of special Lagrangian objects, their relations to the examples constructed in the paper and some similarities and differences between minimal surfaces in \(S^3\) and minimal Legendrian surfaces in \(S^5\). At the end a possible approach to construct more general special Lagrangian cones is presented using the examples in this paper as basic building blocks.

By using these concepts and results, an infinite family of special Lagrangians in \(\mathbb{C}^3\) is constructed, each of which has a toroidal link, obtaining a detailed geometric description of these tori. Then one constructs a one-parameter family of asymptotically conical special submanifolds from any special Lagrangian cones. One discusses some more general singularities of special Lagrangian objects, their relations to the examples constructed in the paper and some similarities and differences between minimal surfaces in \(S^3\) and minimal Legendrian surfaces in \(S^5\). At the end a possible approach to construct more general special Lagrangian cones is presented using the examples in this paper as basic building blocks.

Reviewer: Vasile Oproiu (Iaşi)