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We give a complete characterization of those disk bundles over surfaces which embed as rationally convex strictly pseudoconvex domains in $mathbb{C}^2$. We recall some classical obstructions and prove some deeper ones related to symplectic and contact topology. We explain the close connection to Lagrangian surfaces with isolated singularities and develop techniques for constructing such surfaces. Our proof also gives a complete characterization of Lagrangian surfaces with open Whitney umbrellas, answering a question first posed by Givental in 1986.
The Leray transform and related boundary operators are studied for a class of convex Reinhardt domains in $mathbb C^2$. Our class is self-dual; it contains some domains with less than $C^2$-smooth boundary and also some domains with smooth boundary a
We study the homeomorphic extension of biholomorphisms between convex domains in $mathbb C^d$ without boundary regularity and boundedness assumptions. Our approach relies on methods from coarse geometry, namely the correspondence between the Gromov b
The Riemannian product of two hyperbolic planes of constant Gaussian curvature -1 has a natural Kahler structure. In fact, it can be identified with the complex hyperbolic quadric of complex dimension two. In this paper we study Lagrangian surfaces i
We obtain explicit bounds on the difference between local and global Kobayashi distances in a domain of $mathbb C^n$ as the points go toward a boundary point with appropriate geometric properties. We use this for the global comparison of various inva
These are notes for the CIME school on Complex non-Kahler geometry from July 9th to July 13th of 2018 in Cetraro, Italy. It is an overview of different properties of a class of non-Kahler compact complex manifolds called LVMB manifolds, obtained as t