The Ptolemaean Inequality in the closure of complex hyperbolic plane
We prove Ptolemaean Inequality and Ptolemaeus' Theorem in the closure complex hyperbolic plane endowed with the Cygan metric.
math.MG↗
arXiv subjects
Ioannis D. Platis
Publications and source records attributed to Ioannis D. Platis.
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↗
Modulus of surface families and the radial stretch in the Heisenberg group
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.
math.CV↗
Quasiconformal mappings on the Heisenberg group: An overview
We present a brief overview of the Korányi-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 problems.
math.DG↗
Pseudoconformal structures and the example of Falbel's Cross--Ratio variety
We introduce pseudoconformal 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 ${\rm CR}$ structure. A special case is that of a strictly pseudoconformal structure: in this case, the two complex operators are also opposite in a 2-dimensional subbundle which is complementary to the ${\rm CR}$ structure. A non trivial example of a manifold endowed with a (strictly) pseudoconformal structure is Falbel's cross--ratio variety $\mathfrak{X}$; this variety is isomorphic to the ${\rm PU}(2,1)$ configuration space of quadruples of pairwise distinct points in $S^3$. We first prove that there are two complex structures that appear naturally in $\mathfrak{X}$; these give $\mathfrak{X}$ a pseudoconformal structure which coincides with its well known ${\rm CR}$ structure. Using a non trivial involution of $\mathfrak{X}$ we then prove that $\mathfrak{X}$ is a strictly pseudoconformal manifold. The geometric meaning of this involution as well as its interconnections with the $\rm{CR}$ and complex structures of $\mathfrak{X}$ are also studied here in detail.
math.DG↗
The Ptolemaean Inequality in H--type groups
We prove the Ptolemaean Inequality and the Theorem of Ptolemaeus in the setting of $H$--type groups of Iwasawa--type.
math.MG↗
Straight ruled surfaces in the Heisenberg group
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 plane.
math.DG↗
Cross ratios and the Ptolemaean inequality in boundaries of symmetric spaces of rank 1
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.
math.DG↗
Complex cross--ratios and the Ptolemaean inequality
We use Korányi--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.
math.DG↗