Constructing special Lagrangian m-folds in C^m by evolving quadrics

TL;DR

This paper develops a method to construct explicit special Lagrangian submanifolds in complex Euclidean spaces, focusing on those fibred by quadrics, to better understand singularities relevant to Mirror Symmetry in Calabi-Yau manifolds.

Contribution

It introduces a new construction technique for special Lagrangian m-folds fibred by quadrics, expanding the class of explicit examples and laying groundwork for singularity analysis.

Findings

Constructed special Lagrangian m-folds fibred by quadrics in C^m.

Examples include cones on S^a x S^b x S^1, modeling singularities.

Some examples match previously known constructions by Lawlor and Harvey.

Abstract

This is the second in a series of papers constructing explicit examples of special Lagrangian submanifolds in C^m. The first paper was math.DG/0008021, which studied special Lagrangian m-folds with large symmetry groups. The third is math.DG/0010036, which uses ideas from this paper to construct families of special Lagrangian 3-folds in C^3. This paper describes a construction of special Lagrangian m-folds in C^m which are fibred by (m-1)-submanifolds which are quadrics in Lagrangian planes R^m in C^m. Generically they have only discrete symmetry groups. Some of our examples have been previously constructed by Lawlor and Harvey, using different methods. The principal motivation for these papers is to lay the foundations for the study of singularities of compact special Lagrangian m-folds in Calabi-Yau m-folds. Understanding such singularities will be important in resolving the SYZ conjecture on Mirror Symmetry of Calabi-Yau 3-folds. The special Lagrangian m-folds in C^m we construct here include many cones on S^a x S^b x S^1 for a+b=m-2, which are local models for singularities of special Lagrangian m-folds in Calabi-Yau m-folds.