Gluing constructions of special Lagrangian cones

Gluing constructions of special Lagrangian cones

Tue, 26/05/2009
15:45 		Nicos Kapouleas (US) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Gluing constructions of special Lagrangian cones
I will survey the recent work of Haskins and myself constructing new special Lagrangian cones in $ {\mathbb C}^n $ for all $ n\ge3 $ by gluing methods. The link (intersection with the unit sphere $ {\cal S}^{2n-1} $) of a special Lagrangian cone is a special Legendrian $ (n-1) $-submanifold. I will start by reviewing the geometry of the building blocks used. They are rotationally invariant under the action of $ SO(p)\times SO(q) $ ($ p+q=n $) special Legendrian $ (n-1) $-submanifolds of $ {\cal S}^{2n-1} $. These we fuse (when $ p=1 $, $ p=q $) to obtain more complicated topologies. The submanifolds obtained are perturbed to satisfy the special Legendrian condition (and their cones therefore the special Lagrangian condition) by solving the relevant PDE. This involves understanding the linearized operator and its small eigenvalues, and also ensuring appropriate decay for the solutions.