We study a geometric property of the boundary on Hartogs domains which can be used to find upper and lower bounds for the Diederich–Fornæ ss index. Using this property, we are able to show that under some reasonable hypotheses on the set of weakly pseudoconvex points, the Diederich–Fornæss index for a Hartogs domain is equal to one if and only if the domain admits a family of good vector fields in the sense of Boas and Straube. We also study the analogous problem for a Stein neighborhood basis and show that, under the same hypotheses, if the Diederich–Fornæss index for a Hartogs domain is equal to one, then the domain admits a Stein neighborhood basis.

We show that any subgroup of a finitely generated virtually abelian group $G$ grows rationally relative to $G$, that the set of right cosets of any subgroup of $G$ grows rationally, and that the set of conjugacy classes of $G$ grows rationally. These results hold regardless of the choice of finite weighted generating set for $G$.

In this paper, we classify the hypersurfaces in ${\mathbb{S}}^n\times \mathbb{R}$ and ${\mathbb{H}}^n\times \mathbb{R}$, $n\ne 3$, with $g$ distinct constant principal curvatures, $g\in \{1,2,3\}$, where ${\mathbb{S}}^n$ and ${\mathbb{H}}^n$ denote the sphere and hyperbolic space of dimension $n$, respectively. We prove that such hypersurfaces are isoparametric in those spaces. Furthermore, we find a necessary and sufficient condition for an isoparametric hypersurface in ${\mathbb{S}}^n\times \mathbb{R}$ and ${\mathbb{H}}^n\times \mathbb{R}$ with flat normal bundle when regarded as submanifolds with codimension two of the underlying flat spaces ${\mathbb{R}}^{n+2}\supset {\mathbb{S}}^n\times \mathbb{R}$ and ${\mathbb{L}}^{n+2}\supset {\mathbb{H}}^n\times \mathbb{R}$, having constant principal curvatures.

It is known that there exist Calabi–Yau structures on the complexifications of symmetric spaces of compact type. In this paper, we first construct explicit complete Ricci-flat Kaehler metrics (which give Calabi–Yau structures) for complexified symmetric spaces of arbitrary rank in terms of the Schwarz’s theorem. We consider the case where the Calabi–Yau structure arises from the generalized Stenzel metric. In the complexified symmetric spaces equipped with such a Calabi–Yau structure, we give constructions of special Lagrangian submanifolds of any given phase which are invariant under the actions of symmetric subgroups of the isometry group of the original symmetric space of compact type.

It is shown by construction that if $1<p_1<p_2<\infty$, then there exists a sequence of points in the open unit disk that is a zero set for ${\ell }_A^{p_2}$ but not for ${\ell }_A^{p_1}$. The proof utilizes a zero set criterion for ${\ell }_A^p$ based on a notion of $p$-inner functions.

In this paper, we directly show the known equivalence of simple factor dressings of extended frames and the classical Bianchi–Bäcklund transformations for constant mean curvature surfaces. In doing so, we show how the parameters of classical Bianchi–Bäcklund transformations can be incorporated into the simple factor dressings method.

Let $F$ be a free group of positive, finite rank and let $\Phi \in Aut\left(F\right)$ be a polynomial-growth automorphism. Then $F{⋊}_{\Phi }\mathbb{Z}$ is strongly thick of order $\eta$, where $\eta$ is the rate of polynomial growth of $\varphi$. This fact is implicit in work of Macura, whose results predate the notion of thickness. Therefore, in this note, we make the relationship between polynomial growth of and thickness explicit. Our result combines with a result independently due to Dahmani–Li, Gautero–Lustig, and Ghosh to show that free-by-cyclic groups admit relatively hyperbolic structures with thick peripheral subgroups.