Bernhard Lamel

Regularity and convergence theorems for CR maps

A recurring theme in CR geometry is that maps between CR manifolds are often much more regular than they have any right to be: under suitable nondegeneracy assumptions on the source and the target, formal maps converge, and smooth maps are real-analytic. This minicourse is an introduction to this circle of ideas, with the aim of presenting the techniques behind recent joint work with Nordine Mir, where such theorems are obtained for CR maps of arbitrary codimension.

We begin with an introduction to the regularity and convergence problem, the basic examples and obstructions, and a proof of the convergence proposition for formal maps into strongly pseudoconvex CR manifolds. We then turn to the smooth category and prove a regularity theorem for CR maps of positive codimension; the main tools here are almost analytic extensions and an adapted implicit function theorem. The second half of the course develops the formalism of $\bar\partial_b$-closed modules associated to a CR map, which we then use to construct formal varieties in the image of a divergent (or non-smooth) map; this construction is at the heart of the general dichotomy theorems and explains in what sense the positive results are optimal.

Amar Deep Sarkar

Geometric Analysis of Complex Domains: Gromov Hyperbolicity, Visibility, and Boundary Type

Abstract TBA