Abstract: A Hamiltonian isotopy class of positive Lagrangians in an almost Calabi-Yau manifold admits a natural Riemannian metric. This metric has a Levi-Civita connection, and hence, it gives rise to a notion of geodesics. The geodesic equation is fully non-linear degenerate elliptic, and in general, it is yet unknown whether the initial value problem and boundary problem are well-posed. However, results on the existence of geodesics could shed new light on special Lagrangians, mirror symmetry and the strong Arnold conjecture.
As it turns out, if there is a "big enough" Lie group acting on the manifold, and if we restrict the above discussion to Lagrangians invariant under the action, the initial value problem and boundary problem have unique solutions.
The funny words will be explained in the talk, and examples will be given. Joint work with Jake Solomon.