Some properties of $h$-extendible domains in $\mathbb C^{n+1}$

The purpose of this article is twofold. The first aim is to characterize $h$-extendibility of smoothly bounded pseudoconvex domains in $\mathbb C^{n+1}$ by their noncompact automorphism groups. Our second goal is to show that if the squeezing function tends to $1$ at an $h$-extendible boundary point of a smooth pseudoconvex domain in $\mathbb C^{n+1}$, then this point must be strongly pseudoconvex.