# Regularity of CR maps into uniformly pseudoconvex hypersurfaces and applications to proper holomorphic maps

## TL;DR

This paper investigates the regularity of CR maps into pseudoconvex manifolds with Levi foliations, establishing conditions for smoothness and applying results to boundary regularity of proper holomorphic maps.

## Contribution

It introduces an invariant that determines when CR maps are smooth or constrained, and proves generic smoothness of certain CR maps between pseudoconvex hypersurfaces.

## Key findings

- CR maps are either generically smooth or highly constrained.
- Sufficient regularity implies generic smoothness of CR transversal maps.
- Boundary regularity of proper holomorphic maps into symmetric domains is established.

## Abstract

We study regularity properties of CR maps in positive codimension valued in pseudoconvex manifolds which carry a nontrivial Levi foliation. We introduce an invariant which can be used to deduce that any sufficiently regular CR map from a minimal manifold into such a foliated target is either generically smooth or geometrically highly constrained, and to show generic smoothness of sufficiently regular CR transversal CR maps between pseudoconvex hypersurfaces. As an application, we discuss boundary regularity of proper holomorphic maps into bounded symmetric domains.