Visibility metrics on the infinity boundary of the complex hyperbolic plane

Limiting spherical and horospherical metrics an the infinity boundary of the complex hyperbolic plane are constructed. It is proved that the limiting spherical metric, which automatically is the Carnot–Carathéodory metric, is also a visibility metric, i.e., it belongs to a canonical class of metrics on the infinity boundary. Bibliography: 6 titles.     

Rigidity of certain finite group actions on the complex projective plane

Partially supported by NSERC grant A4000     

Holomorphic helix of proper order 3 on a complex hyperbolic plane

With the aid of the natural fibration on a complex hyperbolic plane we study holomorphic helices of proper order 3 and show that they are not bounded if their second curvature is sufficiently large.     

Real hypersurfaces with constant principal curvatures in the complex hyperbolic plane

We classify real hypersurfaces with constant principal curvatures in the complex hyperbolic plane. It follows from this classification that all of them are open parts of homogeneous ones.

     

Polar actions on Hilbert space

An isometricH-action on a Riemannian manifoldX is calledpolar if there exists a closed submanifoldS ofX that meets everyH-orbit and always meets orbits orthogonally (S is called a section). LetG be a compact Lie group equipped with a biinvariant metric,H a closed subgroup ofG ×G, and letH act onG isometrically by (h ₁,h ₂) ·x = h ₁ xh ₂ ⁻¹ · LetP(G, H) denote the group ofH ¹-pathsg: [0, 1] →G such that (g(0),g (1)) ∈H, and letP(G, H) act on the Hilbert spaceV = H ⁰([0, 1], g) isometrically byg * u = gug ⁻¹ −g′g ⁻¹. We prove that if the action ofH onG is polar with a flat section then the action ofP(G, H) onV is polar. Principal orbits of polar actions onV are isoparametric submanifolds ofV and are infinite-dimensional generalized real or complex flag manifolds. We also note that the adjoint actions of affine Kac-Moody groups and the isotropy action corresponding to an involution of an affine Kac-Moody group are special examples ofP(G, H)-actions for suitable choice ofH andG. Work supported partially by NSF Grant DMS 8903237 and by The Max-Planck-Institut für Mathematik in Bonn.     

Polar actions on Damek-Ricci spaces

A proper isometric Lie group action on a Riemannian manifold is called polar if there exists a closed connected submanifold which meets all orbits orthogonally. In this article we study polar actions on Damek-Ricci spaces. We prove criteria for isometric actions on Damek-Ricci spaces to be polar, find examples and give some partial classifications of polar actions on Damek-Ricci spaces. In particular, we show that non-trivial polar actions exist on all Damek-Ricci spaces.     

Isometric Lagrangian immersion of horocycles of the hyperbolic plane in complex space

We prove that there exists an isometric Lagrangian immersion of a horocycle of the hyperbolic plane in the complex space ?², and there exists an isometric Lagrangian immersion of a horoball of hyperbolic (Lobachevski) space H ³ in the complex space ?³.     

Percolation in the hyperbolic plane

We study percolation in the hyperbolic plane and on regular tilings in the hyperbolic plane. The processes discussed include Bernoulli site and bond percolation on planar hyperbolic graphs, invariant dependent percolations on such graphs, and Poisson-Voronoi-Bernoulli percolation. We prove the existence of three distinct nonempty phases for the Bernoulli processes. In the first phase, , there are no unbounded clusters, but there is a unique infinite cluster for the dual process. In the second phase, , there are infinitely many unbounded clusters for the process and for the dual process. In the third phase, , there is a unique unbounded cluster, and all the clusters of the dual process are bounded. We also study the dependence of in the Poisson-Voronoi-Bernoulli percolation process on the intensity of the underlying Poisson process.

     

Coordinates in the hyperbolic plane

Abstract This note shows that in the hyperbolic plane three kinds of coordinates are possible. Keywords: Hyperbolic plane, Quasi-regular quadrangle, Pseudo-parallelogram, Hypercycle, Hypocycle Mathematics Subject Classification (2000): 20N05     

Bounding surface actions on hyperbolic spaces

Given an isometric action of the fundamental group of a closed orientable surface on a δ-hyperbolic space, we find a standard generating set whose translation distances are bounded above in terms of the hyperbolicity constant δ, the genus of the surface, and the injectivity radius of the action, which we assume to be strictly positive.     

Lattice actions on the plane revisited

We study the action of a lattice Γ in the group G = SL(2, R) on the plane. We obtain a formula which simultaneously describes visits of an orbit Γu to either a fixed ball, or an expanding or contracting family of annuli. We also discuss the ‘shrinking target problem’. Our results are valid for an explicitly described set of initial points: all \({{\bf u} \in {\bf R}^2}\) in the case of a cocompact lattice, and all u satisfying certain diophantine conditions in case \({\Gamma = {\rm SL}(2, \mathbb {Z})}\) . The proofs combine the method of Ledrappier with effective equidistribution results for the horocycle flow on \({\Gamma {\backslash} G}\) due to Burger, Strömbergsson, Forni and Flaminio.     

Polar actions on compact Euclidean hypersurfaces

Given an isometric immersion of a compact Riemannian manifold of dimension n ≥ 3 into Euclidean space of dimension n + 1, we prove that the identity component Iso ⁰(M ⁿ ) of the isometry group Iso(M ⁿ ) of M ⁿ admits an orthogonal representation such that for every . If G is a closed connected subgroup of Iso(M ⁿ ) acting polarly on M ⁿ , we prove that Φ(G) acts polarly on , and we obtain that f(M ⁿ ) is given as Φ(G)(L), where L is a hypersurface of a section which is invariant under the Weyl group of the Φ(G)-action. We also find several sufficient conditions for such an f to be a rotation hypersurface. Finally, we show that compact Euclidean rotation hypersurfaces of dimension n ≥ 3 are characterized by their underlying warped product structure.      

On hyperbolic once-punctured-torus bundles III: Comparing two tessellations of the complex plane

To each once-punctured-torus bundle, Tφ, over the circle with pseudo-Anosov monodromy φ, there are associated two tessellations of the complex plane: one, Δ(φ), is (the projection from ∞ of) the triangulation of a horosphere at ∞ induced by the canonical decomposition into ideal tetrahedra, and the other, CW(φ), is a fractal tessellation given by the Cannon-Thurston map of the fiber group switching back and forth between gray and white each time it passes through ∞. In this paper, we fully describe the relation between Δ(φ) and CW(φ).     

Polar manifolds and actions

A group action is called polar if there exists an immersed submanifold (a section) which intersects all orbits orthogonally. We show how to construct a manifold admitting a polar group action by prescribing its isotropy groups along a fundamental domain in the section. This generalizes a classical construction for cohomogeneity-one manifolds.We give many examples showing the richness of this class of group actions and relate the topology of the section to the topology of the manifold.     

Projective-type axioms for the hyperbolic plane

It is shown that the Laws of Pappus and Desargues may replace the Axiom of Projectivities in Menger's development of hyperbolic geometry from axioms of alignment.     

Support function and hyperbolic plane

Minkowskis theorem _C(cosh d(o, ) – kS) ds = 0 in the hyperbolic plane (Kleins model) for smoothly bounded horocyclic convex bodies K with outer unit normal vector u and curvature |k| 1 of C K with arclength s where S <sinh d(o, ) grad d(o, ), u> motivates the introduction of a hyperbolic support function H of K. Hereby H() d(l(), D⁺()) is the distance of the K-supporting distance curve D⁺() from the line l() through the origin o with the direction angle . – The paper deals with the representation of C, s and k by H including extremal cases and an application of Minkowskis theorem to the characterization of circles by inequalities for their hyperbolic support function.