Ioannis D. PlatisUniversity of Crete | UOC · Department of Mathematics and Applied Mathematics
Ioannis D. Platis
PhD
About
45
Publications
4,166
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Introduction
I teach Mathematics in the University of Crete. My interests vary from complex analysis to differential geometry. In particular, I work on hyperbolic and complex hyperbolic geometry, quasiconformal mappings of the complex plane and Teichmuller theory. For quite a while, I also work on sub-Riemannian geometry and quasiconformal mappings of the Heisenberg group, a really fascinating field to work in.
Additional affiliations
January 2015 - February 2015
February 2014 - March 2014
December 2013 - present
Education
September 1994 - December 1999
University of Crete, Heraklion
Field of study
- Mathematics
September 1992 - June 1994
University of Crete, Heraklion Crete
Field of study
- Mathematics
September 1984 - June 1988
Publications
Publications (45)
Let HKn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{H}}^n_{{\mathbb K}}$$\end{document} denote the symmetric space of rank-1 and of non-compact type and let...
The modulus method introduced by H. Grötzsch yields bounds for a mean distortion functional of quasiconformal maps between two annuli mapping the respective boundary compo-nents onto each other. P. P. Belinski˘ ı studied these inequalities in the plane and identified the family of all minimisers. Beyond the Euclidean framework, a Grötzsch–Belinski˘...
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...
We present a brief overview of the Kor\'anyi-Reimann theory of quasiconformal
mappings on the Heisenberg group stressing on the analogies as well as on the
differences between the Heisenberg group case and the classical two-dimensional
case. We examine the extensions of the theory to more general spaces and we
state some known results and open prob...
The Korányi ellipsoidal ring E of radii B and A, 0 < B < A, is defined as the image of the Korányi spherical ring of the same radii and centred at the origin via a linear contact map L in the Heisenberg group. If K ≥ 1 is the maximal distortion of L then we prove that the modulus of E is equal to mod(E) = 3 8 K 2 + 1 K 2 + 1 4 π 2 (log(A/B)) 3 .
Let H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathfrak H}}$$\end{document} be the first Heisenberg group equipped with the Korányi metric d. We prove that the...
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 K\"ahler structure: this structure...
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.
We show that metric bisectors with respect to the Kor\'anyi metric in the Heisenberg group are spinal spheres and vice versa. We also calculate explicitly their horizontal mean curvature.
Let $\mathfrak{H}$ be the first Heisenberg group equipped with the Kor\'anyi metric $d$. We prove that the equilateral dimension of $\mathfrak{H}$ is 4.
In this short note we endow the right half plane with a complete metric with non constant negative curvature. The geodesics here are hyperbolas and vertical straight lines. This procedure can be generalised to all non compact symmetric spaces of rank one; details will appear elsewhere.
We study X-valued CR functions of Theodoresco class B1 on the Heisenberg group H1=R×C equipped with the non-embeddable CR structure discovered by Nirenberg [On a question of Hans Lewy. Russian Math Surveys. 1974;29:251–262] where X is an arbitrary complex Fréchet space. If L¯φ≡L¯+φ∂/∂x is Nirenberg's perturbation of Lewy's operator L¯=∂/∂w¯−iw∂/∂x,...
We show that an open subset \({\mathfrak {F}}_4''\) of the \(\mathrm{PU}(2,1)\) configuration space of four points in \(S^3\) is in bijection with an open subset 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...
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.
We prove that the ${\rm PU}(2,1)$ configuration space ${\mathfrak F}_4$ of four points in $S^3$ is in bijection with ${\mathfrak H}^{\star}\times\mathbb{R}_{>0}$, where ${\mathfrak H}^\star$ is the hyperbolic Heisenberg group. The latter is a Sasakian manifold and therefore ${\mathfrak H}^\star\times\mathbb{R}_{>0}$ is K\"ahler.
The Kor\'anyi ellipsoidal ring $\mathcal{E}$ of radii $B$ and $A$, $0<B<A$, is defined as the image of the Kor\'anyi spherical ring of the same radii and centred at the origin via a linear contact map $L$ in the Heisenberg group. If $K\ge 1$ is the maximal distortion of $L$ then we prove that the modulus of $\mathcal{E}$ is equal to $$ {\rm mod}(\m...
We prove that the configuration space of equidistant triples on the Heisenberg group equipped with the Kor\'anyi metric, is isomorphic to a hypersurface of $\mathbb{R}^3$.
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 F\"urstenberg boundary $\mathbb{F}(X)$ 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 ${\...
We prove that the configuration space of equidistant triples on the Heisenberg group equipped with the Korányi metric, is isomorphic to a hypersurface of R 3 .
We prove Ptolemaean Inequality and Ptolemaeus' Theorem in the closure complex hyperbolic plane endowed with the Cygan metric.
We introduce paired (Formula presented.) structures on 4-dimensional manifolds and study their properties. Such structures are arising from two different complex operators which agree in a 2-dimensional subbundle of the tangent bundle; this subbundle thus forms a codimension 2 (Formula presented.) structure. A special case is that of a strictly pai...
We introduce pseudoconformal structures on 4--dimensional manifolds and study
their properties. Such structures are arising from two complex operators which
commute in a 2--dimensional subbundle of the tangent bundle; this subbundle
thus forms a codimension 2 $\C\R$ structure. A non trivial example of a
manifold endowed with a pseudoconformal struc...
We generalise a result of Garofalo and Pauls: a horizontally minimal smooth
surface embedded in the Heisenberg group is locally a (straight) ruled surface,
i.e. it consists of straight lines tangent to a horizontal vector field along a
smooth curve. We show additionally that any horizontally minimal surface is
locally contactomorphic to the complex...
We develop a modulus method for surface families inside a domain in the Heisenberg group and we prove that the stretch map between two Heisenberg spherical rings is a minimiser for the mean distortion among the class of contact quasiconformal maps between these rings which satisfy certain boundary conditions.
We develop a modulus method for surface families inside a domain in the
Heisenberg group and we prove that the stretch map between two Heisenberg
spherical rings is a minimiser for the mean distortion among the class of
contact quasiconformal maps between these rings which satisfy certain boundary
conditions.
We prove the Ptolemaean Inequality and the Theorem of Ptolemaeus
in the setting of H–type groups of Iwasawa–type.
We propose a method by modulus of curve families to identify extremal qua-siconformal mappings in the Heisenberg group. This approach allows to study minimizers not only for the maximal distortion but also for a mean distortion functional, where the candidate for the extremal map is not required to have constant distortion. As a counterpart of a cl...
We use generalised cross--ratios to prove the Ptolemaean inequality and the
Theorem of Ptolemaeus in the setting of the boundary of symmetric Riemannian
spaces of rank 1 and of negative curvature.
We use Kor\'anyi--Reimann complex cross--ratios to prove the Ptolemaean
inequality and the Theorem of Ptolemaeus in the setting of the boundary of
complex hyperbolic space and the first Heisenberg group.
We develop a method using the modulus of curve families to study minimisation problems for the mean distortion functional
in the class of finite distortion homeomorphisms. We apply our method to prove extremality of the spiral-stretch mappings
defined on annuli in the complex plane. This generalises results of Gutlyanskiĭ and Martio [12] and Strebe...
A complex hyperbolic quasi-Fuchsian group is a discrete, faithful, type preserving and geometrically finite representation of a surface group as a subgroup of the group of holomorphic isometries of complex hyperbolic space. Such groups are direct complex hyperbolic generali-sations of quasi-Fuchsian groups in three dimensional (real) hyperbolic geo...
Riemann surfaces is a thriving area of mathematics with applications to hyperbolic geometry, complex analysis, fractal geometry, conformal dynamics, discrete groups, geometric group theory, algebraic curves and their moduli, various kinds of deformation theory, coding, thermodynamic formalism, and topology of three-dimensional manifolds. This colle...
Complex hyperbolic packs are 3-hypersurfaces of complex hyperbolic plane H 2 C which may be considered as dual to the well known bisectors. In this article we study the geometric aspects associated to packs.
Falbel has shown that four pairwise distinct points on the boundary of a complex hyperbolic 2-space are completely determined, up to conjugation in PU(2,1), by three complex cross-ratios satisfying two real equations. We give global geometrical coordinates on the resulting variety.
The configuration space of four points on the standard CR 3-sphere up to CR-automorphisms is a real four dimensional variety.
We prove the existence of natural complex and CR structures on this space.
We study quakebend deformations in complex hyperbolic quasi-Fuchsian space Q C (Σ) of a closed surface Σ of genus g > 1, that is the space of discrete, faithful, totally loxodromic and geometrically finite representations of the fundamental group of Σ into the group of isometries of complex hyperbolic space. Emanating from an R−Fuchsian point ρ ∈ Q...
Let Σ be a closed, orientable surface of genus g. It is known that the representation variety of π1(Σ) has 2g−3 components of (real) dimension 16g−16 and two components of dimension 8g−6. Of special interest are the totally loxodromic, faithful (that is quasi-Fuchsian) representations. In this paper we give global real analytic coordinates on a sub...
Let π be the fundamental group of a closed surface Σ of genus g > 1. One of the fundamental problems in complex hyperbolic geometry is to find all discrete, faithful, geometrically finite and purely loxodromic representations of π into SU(2, 1), (the triple cover of) the group of holomorphic isometries of H<sup>2</sup><sub>C</sub>. In particular, g...
Let T be a cross product of n Teichmüller spaces of Fuchsian groups, n > 1. From the properties of Kobayashi metric and from the Royden-Gardiner theorem, T is a complete hyperbolic manifold. Each two distinct points of T can be joined by a hyperbolic geodesic segment, which is not in general unique. But when T is finite dimensional or infinite dime...
We study the complex symplectic geometry of the space QF(S) of quasi-Fuchsian structures of a compact orientable surface S of genus g > 1. We prove that QF(S) is a complex symplectic manifold. The complex symplectic structure is the complexification of the Weil–Petersson symplectic structure of Teichmller space and is described in terms which look...
We study the complex symplectic geometry of the space Q(S) of the quasifuchsian structures of a closed Riemannian surface S of genus g>1. We prove that this space is a flat complex symplectic manifold and we describe the hamiltonian nature of the quasifuchsian bending vector fields.
Questions
Questions (4)
Η^2ΧR admits a Lie group structure-what I call Affine-Rotational, or Hyperbolic Heisenberg group. This has a beautiful Sasakian structure. Perhaps it will help to check my project.
According to Capogna, Danielli, Tyson & Pauls (An Introduction to the Heisenberg group and the sub Riemannian isoperimetric problem), the isoperimetric best constant in the Heisenberg group is conjectured (by P. Pansu) to be 3^{3/4}/4\sqrt{\pi} (attained for the bubble set). However, from an unpublished paper of Capogna (Isoperimetric inequalities in the Heisenberg group and in the plane), it follows from the formulae therein that the best constant is 3^{3/4}/4^{5/4}\sqrt{\pi}. Which is the correct one?
Is there a publication of a natural definition of octonionic cross--ratios? I have had trouble defining them due to non associativity of octonions. Can this be by--passed?
To prove that a CAT(0) space is Ptolemaean is relatively easy, and the converse does not always hold (obviously, e.g. the Heisenberg group with the Koranyi is Ptolemaean). There must be some other equally intrinsic property, such that in addition to the Ptolemaean property turns a space to a CAT(0) space. Might this be as elementary as well? Any ideas are welcome.
