We prove a McShane-type identity: a series, expressed in terms of geodesic lengths, that sums to 2π for any closed hyperbolic surface with one distinguished point. To do so, we prove a generalized Birman-Series theorem showing that the set of complete geodesics on a hyperbolic surface with large cone angles is sparse.

We apply the mean curvature flow to deform symplectomorphisms of CP^n. In particular, we prove that, for each dimension n, there exists a constant , explicitly computable, such that anypinched symplectomorphism of CP^n is symplectically isotopic to a biholomorphic isometry.

We prove that the number of critical points of a Li–Tam Green’s function on a complete open Riemannian surface of finite type admits a topological upper bound, given by the first Betti number of the surface. In higher dimensions, we show that there are no topological upper bounds on the number of critical points by constructing, for each nonnegative integer N, a Riemannian manifold diffeomorphic to Rn (n > 3) whose minimal Green’s function has at least N non-degenerate critical points. Variations on the method of proof of the latter result yield contractible n-manifolds whose minimal Green’s functions have level sets diffeomorphic to any fixed codimension 1 compact submanifold of R^n.

We exhibit a transformation taking special Lagrangian submanifolds of a Calabi-Yau together with local systems to vector bundles over the mirror manifold with connections obeying deformed Hermitian-Yang-Mills equations. That is, the transformation relates supersymmetric A- and Bcycles. In this paper, we assume that the mirror pair are dual torus fibrations with flat tori and that the A-cycle is a section. We also show that this transformation preserves the (holomorphic) Chern-Simons functional for all connections. Furthermore, on corresponding moduli spaces of supersymmetric cycles it identifies the graded tangent spaces and the holomorphic m-forms. In particular, we verify Vafa’s mirror conjecture with bundles in this special case.

We make a conjecture about mean curvature flow of Lagrangian submanifolds of Calabi-Yau manifolds, expanding on\cite {Th}. We give new results about the stability condition, and propose a Jordan-Hlder-type decomposition of (special) Lagrangians. The main results are the uniqueness of special Lagrangians in hamiltonian deformation classes of Lagrangians, under mild conditions, and a proof of the conjecture in some cases with symmetry: mean curvature flow converging to Shapere-Vafa's examples of SLags.