A non-strictly pseudoconvex domain for which the squeezing function tends to one towards the boundary

TL;DR

This paper demonstrates that the previous link between the squeezing function approaching one and strict pseudoconvexity in convex domains does not hold if the boundary smoothness is only $C^2$, providing a counterexample.

Contribution

It constructs a non-strictly pseudoconvex domain with $C^2$ boundary where the squeezing function tends to one near the boundary, challenging prior assumptions.

Findings

Counterexample to the previous theorem

Squeezing function can tend to one without strict pseudoconvexity

Boundary smoothness level affects the relation between squeezing function and pseudoconvexity

Abstract

In recent work by Zimmer it was proved that if $\Omega\subset\mathbb C^n$ is a bounded convex domain with $C^\infty$-smooth boundary, then $\Omega$ is strictly pseudoconvex provided that the squeezing function approaches one as one approaches the boundary. We show that this result fails if $\Omega$ is only assumed to be $C^2$-smooth.