We study the symplectic structure of the holomorphic coadjoint orbits, generalizing a theorem of McDuff on the symplectic structure of Hermitian symmetric spaces of noncompact type.

In this paper, we study the behavior of Ricci-flat K¨ahler metrics on Calabi–Yau manifolds under algebraic geometric surgeries: extremal transitions or flops. We prove a version of Candelas and de la Ossa’s conjecture: Ricci-flat Calabi–Yau manifolds related by extremal transitions and flops can be connected by a path consisting of continuous families of Ricci-flat Calabi–Yau manifolds and a compact metric space in the Gromov–Hausdorff topology. In an essential step of the proof of our main result, the convergence of Ricci-flat K¨ahler metrics on Calabi–Yau manifolds along a smoothing is established, which can be of independent interest.

Given a compact orientable surface, we determine a complete set of relations for a function defined on the set of all homotopy classes of simple loops to be a geometric intersection number function. As a consequence, Thurston’s space of measured laminations and Thurston’s compactification of the Teichm¨uller space are described by a set of explicit equations. These equations are polynomials in the max-plus semi-ring structure on the real numbers. It shows that Thurston’s theory of measured laminations is within the domain of tropical geometry.

We show that the interior of the convex core of a quasifuchsian punctured-torus group admits an ideal decomposition (usually an infinite triangulation) which is canonical in two very different senses: in a combinatorial sense via the pleating invariants, and in a geometric sense via an Epstein-Penner convex hull construction in Minkowski space. This result re-proves the Pleating Lamination Theorem for quasifuchsian punctured-torus groups, and extends to all punctured-torus groups if a strong version of the Pleating Lamination Conjecture is true.

Given a Riemannian manifold and a closed submanifold, we find a geodesic segment with free boundary on the given submanifold. This is a corollary of the min–max theory which we develop in this article for the free boundary variational problem. In particular, we develop a modified Birkhoff curve shortening process to achieve a strong “Colding–Minicozzi” type min–max approximation result.