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Ioannis D. Platis

Publications and source records attributed to Ioannis D. Platis.

At least 19 recordsLinked to original sources
arXiv2026

The ${\rm PSL}(2,\mathbb{R})^2$-configuration space of four points in the torus $S^1\times S^1$

The torus $\mathbb{T}=S^1\times S^1$ appears as the ideal boundary $\partial_\infty AdS^3$ of the three-dimensional anti-de Sitter space $AdS^3$, as well as the Fürstenberg boundary $\mathcal{F}(X)$ of the rank-2 symmetric space $X={\rm SO}_0(2,2)/{\rm SO}(2)\times{\rm SO}(2)$. We introduce cross-ratios on the torus in order to parametrise the ${\rm PSL}(2,\mathbb{R})^2$ configuration space of quadruples of pairwise distinct points in $\mathbb{T}$ and define a natural Möbius structure on $\mathbb{T}$ and therefore on $\mathcal{F}(X)$ and $\partial_\infty AdS^3$ as well.

math.DG↗
arXiv2026

Strictly paired ${\rm CR}$ linear automorphisms of $\R^4$

This paper investigates the algebraic and differential geometric properties of smooth mappings between domains in $\mathbb{C}^2$, classifying them according to the constant ranks of their holomorphic and antiholomorphic derivative components. Particular emphasis is placed on strictly paired CR (SPCR) linear automorphisms of $\mathbb{R}^4$, where both block matrices $A$ and $B$ have rank 1. We characterise the set of SPCR linear automorphisms as a 12-dimensional submanifold of $\mathrm{GL}(2,\mathbb{C})^2$ and analyse its underlying CR structure, demonstrating its geometric and structural rigidity.

math.CV↗
arXiv2026

Modulus of elementary domains in the hyperbolic plane

We study the modulus of curve families inside elementary domains of the hyperbolic plane $\mathbb{H}^1_\mathbb{C}$. We establish exact expressions for the modulus of connecting and separating curve families within a hyperbolic circular annulus. In contrast, for the normal hyperbolic quadrilateral, we construct sharp analytical lower bounds by restricting the metric optimisation certain subfamilies of curves, and we bracket these estimates with upper bounds using Dirichlet energy test functions. Finally, we demonstrate the barriers preventing an explicit, closed-form expression.

math.CV↗
arXiv2026

Horizontal curvatures of surfaces in 3D contact sub-Riemannian Lie groups

In this paper we study horizontal curvatures for surfaces embedded in three-dimensional contact sub-Riemannian Lie groups. Using a Riemannian approximation scheme, we derive explicit formulas for horizontal Gauss curvature, horizontal mean curvature, and symplectic distortion for surfaces embedded in three dimensional Lie groups with a sub-Riemannian structure obtained by a contact form. We focus on two primary examples: the Heisenberg group and the affine-additive group. We classify surfaces of revolution within these groups that exhibit constant horizontal curvatures, often expressing their profiles through elementary or elliptic integrals.

math.DG↗
arXiv2026

On the sub-Riemannian geometry of the quaternionic Heisenberg group

Utilizing the framework of quaternionic contact geometry, we define a sequence of Riemannian metrics $\{g_L\}$ on the quaternionic Heisenberg group $\mathfrak{H}_{\mathbb{H}}$ by rescaling the vertical directions. By analyzing the limit of this sequence, we characterize the Carnot-Carathéodory geodesics and provide the explicit description of the Carnot-Carathéodory distance and spheres in $\mathfrak{H}_{\mathbb{H}}$ . Furthermore, we derive a general formula for the horizontal mean curvature of hypersurfaces.

math.DG↗
arXiv2024

Hyperbolicity of the sub-Riemannian affine-additive group

We consider the affine-additive group as a metric measure space with a canonical left-invariant measure and a left-invariant sub-Riemannian metric. We prove that this metric measure space is locally 4-Ahlfors regular and it is hyperbolic, meaning that it has a non-vanishing 4-capacity at infinity. This implies that the affine-additive group is not quasiconformally equivalent to the Heisenberg group or to the roto-translation group in contrast to the fact that both of these groups are globally contactomorphic to the affine-additive group. Moreover, each quasiregular map, from the Heisenberg group to the affine-additive group must be constant.

math.MG↗
arXiv2023

PCR-Kähler equivalent metrics in the Siegel domain

Let $\mathfrak{H}$ be the Heisenberg group. From the standard CR structure $\mathcal{H}$ of $\mathfrak{H}$ we construct the complex hyperbolic structure of the Siegel domain. Additionally, using the same minimal data for $\mathfrak{H}$, that is, its Sasakian structure, we provide the Siegel domain with yet another Kähler structure: this structure is of unbounded negative sectional curvature, and its complex structure does not commute with the standard complex structure. However, we show that those two Kähler structures are {\rm PCR} Kähler equivalent, that is to say, essentially the same when restricted to $\mathcal{H}$.

math.DG↗
arXiv2023

Half plane geometries

In this paper, we endow the right half plane with warped product metrics. The group of holomorphic isometries of all such metrics is isomorphic to the real additive group. Of our interest are two of those metrics: they have zero and unbounded negative sectional curvature, respectively, and both of them are not complete.

math.DG↗
arXiv2022

Bisectors in the Heisenberg group I

We show that metric bisectors with respect to the Korányi metric in the Heisenberg group are spinal spheres and vice versa. We also calculate explicitly their horizontal mean curvature.

math.DG↗
arXiv2021

A Kähler structure for the ${\rm PU}(2, 1)$ configuration space of four points in $S^3$

We show that an open subset ${\mathfrak F}_4''$ of the ${\rm PU}(2,1)$ configuration space of four points in $S^3$ is in bijection with an open subset of %with a Kähler structure which is inherited from the one of ${\mathfrak H}^{\star}\times\mathbb{R}_{>0}$, where ${\mathfrak H}^\star$ is the affine-rotational group. Since the latter is a Sasakian manifold, the cone ${\mathfrak H}^\star\times\mathbb{R}_{>0}$ is Kähler and thus ${\mathfrak F}_4''$ inherits this Kähler structure.

math.DG↗
arXiv2019

Quasiconformal mappings in the hyperbolic Heisenberg group and a lifting theorem

A study of smooth contact quasiconformal mappings of the hyperbolic Heisenberg group is presented in this paper. Our main result is a Lifting Theorem; according to this, a symplectic quasiconformal mapping of the hyperbolic plane can be lifted to a circles preserving quasiconformal mapping of the hyperbolic Heisenberg group.

math.DG↗
arXiv2015

Modulus of revolution rings in the Heisenberg group

Let $\mathcal{S}$ be a surface of revolution embedded in the Heisenberg group $\mathcal{H}$. A revolution ring $R_{a,b}(\mathcal{S})$, $0<a<b$, is a domain in $\mathcal{H}$ bounded by two dilated images of $\mathcal{S}$, with dilation factors $a$ and $b$, respectively. We prove that if $\mathcal{S}$ is subject to certain geometric conditions, then the modulus of the family $Γ$ of horizontal boundary connecting curves inside $R_{a,b}(\mathcal{S})$ is $$ {\rm Mod}(Γ)=π^2(\log(b/a))^{-3}. $$ Our result applies for many interesting surfaces, e.g., the Korányi metric sphere, the Carnot-Carathéodory metric sphere and the bubble set.

math.MG↗