Deformations of Strongly Pseudoconvex Domains

TL;DR

This paper proves that smoothly bounded, strongly pseudoconvex domains that are diffeomorphic can be smoothly deformed into each other while maintaining strong pseudoconvexity, linking to Kobayashi extremal discs.

Contribution

It establishes a deformation result for strongly pseudoconvex domains, showing their topological equivalence can be realized through smooth deformations preserving pseudoconvexity.

Findings

Diffeomorphic strongly pseudoconvex domains can be smoothly deformed into each other.

All intermediate domains in the deformation remain strongly pseudoconvex.

The result connects to concepts of Kobayashi extremal discs.

Abstract

We show that two smoothly bounded, strongly pseudoconvex domains which are diffeomorphic may be smoothly deformed into each other, with all intermediate domains being strongly pseudoconvex. This result relates to Lempert's ideas about Kobayashi extremal discs, and also has intrinsic interest.