The cylindrical transform

Abstract: A Lagrangian submanifold of a Calabi-Yau manifold is called positive if the restriction to it of the real part of the holomorphic volume form is positive. The space of positive Lagrangians admits a Riemannian metric of non-positive curvature. Understanding the geodesics of the space of positive Lagrangian submanifolds would shed light on questions ranging from the uniqueness and existence of volume minimizing Lagrangian submanifolds to Arnold's nearby Lagrangian conjecture. The geodesic equation is a non-linear degenerate elliptic PDE. I will describe work with A. Yuval on the cylindrical transform, which converts the geodesic equation to a family of non-degenerate elliptic boundary value problems. As a result, we obtain examples of geodesics in arbitrary dimension that are not invariant under any isometries. The talk will be aimed at a broad audience.

differential geometrydynamical systemsgeometric topologysymplectic geometry

Audience: researchers in the topic

Geometry and Dynamics seminar

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