Rationally Convex Domains and Singular Lagrangian Surfaces in $\mathbb{C}^2$

TL;DR

This paper characterizes which disk bundles over surfaces can be embedded as rationally convex strictly pseudoconvex domains in ^2, linking symplectic topology and singular Lagrangian surfaces, and classifies Lagrangian surfaces with specific singularities.

Contribution

It provides a complete characterization of rationally convex domains in ^2 and classifies Lagrangian surfaces with isolated singularities, answering a longstanding question.

Findings

Characterization of disk bundles as rationally convex domains in ^2

Identification of classical and new obstructions related to symplectic topology

Complete classification of Lagrangian surfaces with open Whitney umbrellas

Abstract

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.