A Uniqueness Theorem for Gluing Special Lagrangian Submanifolds

13 March 2012
Yohsuke Imagi
Special Lagrangian submanifolds are area minimizing Lagrangian submanifolds discovered by Harvey and Lawson. There is no obstruction to deforming compact special Lagrangian submanifolds by a theorem of Mclean. It is however difficult to understand singularities of special Lagrangian submanifolds (varifolds). Joyce has studied isolated singularities with multiplicity one smooth tangent cones. Suppose that there exists a compact special Lagrangian submanifold M of dimension three with one point singularity modelled on the Clliford torus cone. We may apply the gluing technique to M by a theorem of Joyce. We obtain then a compact non-singular special Lagrangian submanifold sufficiently close to M as varifolds in Geometric Measure Theory. The main result of this talk is as follows: all special Lagrangian varifolds sufficiently close to M are obtained by the gluing technique. The proof is similar to that of a theorem of Donaldson in the Yang-Mills theory. One first proves an analogue of Uhlenbeck's removable singularities theorem in the Yang-Mills theory. One uses here an idea of a theorem of Simon, who proved the uniqueness of multiplicity one tangent cones of minimal surfaces. One proves next the uniqueness of local models for desingularizing M (see above) using symmetry of the Clifford torus cone. These are the main part of the proof.
  • Algebraic and Symplectic Geometry Seminar