scholarly journals Doubling Property for BiLipschitz Homogeneous Geodesic Surfaces

2010 ◽  
Vol 21 (4) ◽  
pp. 783-806 ◽  
Author(s):  
Enrico Le Donne
Keyword(s):  
Doubling Property ◽  
Homogeneous Geodesic ◽  
Geodesic Surfaces

scholarly journals Embedding Theorems and Area Operators on Bergman Spaces with Doubling Measure

2021 ◽  
Vol 15 (3) ◽  
Author(s):  
Changbao Pang ◽  
Antti Perälä ◽  
Maofa Wang
Keyword(s):  
Borel Measure ◽  
Bergman Spaces ◽  
Embedding Theorem ◽  
Weighted Bergman Spaces ◽  
Embedding Theorems ◽  
Positive Borel Measure ◽  
Doubling Property ◽  
Area Operator ◽  
Special Cases ◽  
Upper Half Plane

AbstractWe establish an embedding theorem for the weighted Bergman spaces induced by a positive Borel measure $$d\omega (y)dx$$ d ω ( y ) d x with the doubling property $$\omega (0,2t)\le C\omega (0,t)$$ ω ( 0 , 2 t ) ≤ C ω ( 0 , t ) . The characterization is given in terms of Carleson squares on the upper half-plane. As special cases, our result covers the standard weights and logarithmic weights. As an application, we also establish the boundedness of the area operator.

scholarly journals Totally geodesic surfaces and homology

2006 ◽  
Vol 6 (3) ◽  
pp. 1413-1428 ◽  
Author(s):  
Jason DeBlois
Keyword(s):  
Totally Geodesic ◽  
Geodesic Surfaces

scholarly journals On the density of Cayley graphs of R.Thompson’s group F in symmetric generators

2021 ◽  
pp. 1-13
Author(s):  
V. S. Guba
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Finite Graph ◽  
Vertex Degree ◽  
Doubling Property ◽  
Generating Set ◽  
Finite Set ◽  
Standard Set ◽  
Thompson’S Group ◽  
Average Vertex

By the density of a finite graph we mean its average vertex degree. For an [Formula: see text]-generated group, the density of its Cayley graph in a given set of generators, is the supremum of densities taken over all its finite subgraphs. It is known that a group with [Formula: see text] generators is amenable if and only if the density of the corresponding Cayley graph equals [Formula: see text]. A famous problem on the amenability of R. Thompson’s group [Formula: see text] is still open. Due to the result of Belk and Brown, it is known that the density of its Cayley graph in the standard set of group generators [Formula: see text], is at least [Formula: see text]. This estimate has not been exceeded so far. For the set of symmetric generators [Formula: see text], where [Formula: see text], the same example only gave an estimate of [Formula: see text]. There was a conjecture that for this generating set equality holds. If so, [Formula: see text] would be non-amenable, and the symmetric generating set would have the doubling property. This would mean that for any finite set [Formula: see text], the inequality [Formula: see text] holds. In this paper, we disprove this conjecture showing that the density of the Cayley graph of [Formula: see text] in symmetric generators [Formula: see text] strictly exceeds [Formula: see text]. Moreover, we show that even larger generating set [Formula: see text] does not have doubling property.

scholarly journals Non-simple geodesics in hyperbolic 3-manifolds

1994 ◽  
Vol 116 (2) ◽  
pp. 339-351
Author(s):  
Kerry N. Jones ◽  
Alan W. Reid
Keyword(s):  
Closed Geodesic ◽  
Closed Geodesics ◽  
Totally Geodesic ◽  
Geodesic Surfaces ◽  
Simple Closed Geodesics

AbstractChinburg and Reid have recently constructed examples of hyperbolic 3-manifolds in which every closed geodesic is simple. These examples are constructed in a highly non-generic way and it is of interest to understand in the general case the geometry of and structure of the set of closed geodesics in hyperbolic 3-manifolds. For hyperbolic 3-manifolds which contain immersed totally geodesic surfaces there are always non-simple closed geodesics. Here we construct examples of manifolds with non-simple closed geodesics and no totally geodesic surfaces.

The Generalized Cuspidal Cohomology Problem

2006 ◽  
Vol 58 (4) ◽  
pp. 673-690 ◽  
Author(s):  
Anneke Bart ◽  
Kevin P. Scannell
Keyword(s):  
Tangent Vector ◽  
Group Cohomology ◽  
Natural Generalization ◽  
Finite Range ◽  
Totally Geodesic ◽  
First Order ◽  
Link Complement ◽  
Cuspidal Cohomology ◽  
Zero Group ◽  
Geodesic Surfaces

AbstractLet Γ ⊂ SO(3, 1) be a lattice. The well known bending deformations, introduced by Thurston and Apanasov, can be used to construct non-trivial curves of representations of Γ into SO(4, 1) when Γ\ℍ3 contains an embedded totally geodesic surface. A tangent vector to such a curve is given by a non-zero group cohomology class in H1(Γ, ℍ41). Our main result generalizes this construction of cohomology to the context of “branched” totally geodesic surfaces. We also consider a natural generalization of the famous cuspidal cohomology problem for the Bianchi groups (to coefficients in non-trivial representations), and perform calculations in a finite range. These calculations lead directly to an interesting example of a link complement in S3 which is not infinitesimally rigid in SO(4, 1). The first order deformations of this link complement are supported on a piecewise totally geodesic 2-complex.

The existence and boundedness of square functions on generalized Orlicz–Campanato spaces

2010 ◽  
Vol 17 (2) ◽  
pp. 405-421
Author(s):  
Songyan Zhang
Keyword(s):  
Single Point ◽  
Positive Function ◽  
Square Functions ◽  
Doubling Property ◽  
Area Function ◽  
Almost Everywhere

Abstract Let T(ƒ) denote the Littlewood–Paley square operators, including the g-function g(ƒ), Luzin area function S(ƒ) and Stein's function , on the generalized Orlicz–Campanato spaces , where Φ is a N-function satisfying the Δ2 condition and φ a positive function satisfying the doubling property. It is proved that if T(ƒ)(x 0) < ∞ for a single point , then T(ƒ)(x) exists almost everywhere in , and T(ƒ) is bounded on the spaces .

scholarly journals Gaussian heat kernel estimates: From functions to forms

2020 ◽  
Vol 2020 (761) ◽  
pp. 25-79
Author(s):  
Thierry Coulhon ◽  
Baptiste Devyver ◽  
Adam Sikora
Keyword(s):  
Riemannian Manifold ◽  
Heat Kernel ◽  
Ricci Curvature ◽  
Compact Riemannian Manifold ◽  
Kernel Estimates ◽  
Doubling Property ◽  
Negative Part ◽  
Heat Kernel Estimates ◽  
Volume Doubling ◽  
Upper Estimate

AbstractOn a complete non-compact Riemannian manifold satisfying the volume doubling property, we give conditions on the negative part of the Ricci curvature that ensure that, unless there are harmonic 1-forms, the Gaussian heat kernel upper estimate on functions transfers to one-forms. These conditions do no entail any constraint on the size of the Ricci curvature, only on its decay at infinity.

scholarly journals Compressing totally geodesic surfaces

2002 ◽  
Vol 118 (3) ◽  
pp. 309-328 ◽  
Author(s):  
Christopher J. Leininger
Keyword(s):  
Totally Geodesic ◽  
Geodesic Surfaces

TOTALLY GEODESIC SURFACES AND QUADRATIC FORMS

2013 ◽  
Vol 22 (13) ◽  
pp. 1350072
Author(s):  
PRADTHANA JAIPONG
Keyword(s):  
Upper Bound ◽  
Quadratic Forms ◽  
Incompressible Surface ◽  
Totally Geodesic ◽  
Geodesic Surfaces ◽  
Figure Eight Knot ◽  
Dehn Fillings

Let M be a compact, connected, irreducible, orientable 3-manifold with torus boundary. A closed, orientable, immersed, incompressible surface F in M with no incompressible annulus joining F and ∂M compresses in at most finitely many Dehn fillings M(α). It is known that there is no universal upper bound on the number of such fillings, independent of the surface, and the figure-eight knot complement is the first example of a manifold where this phenomenon occurs. In this paper, we show that the same behavior of the figure-eight knot complement is shared by other two cusped manifolds.

scholarly journals The Existence of Two Homogeneous Geodesics in Finsler Geometry

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 850
Author(s):  
Zdeněk Dušek
Keyword(s):  
Finsler Geometry ◽  
Finsler Manifold ◽  
Finsler Manifolds ◽  
Homogeneous Geodesics ◽  
Homogeneous Geodesic ◽  
Randers Metrics

The existence of a homogeneous geodesic in homogeneous Finsler manifolds was positively answered in previous papers. However, the result is not optimal. In the present paper, this result is refined and the existence of at least two homogeneous geodesics in any homogeneous Finsler manifold is proved. In a previous paper, examples of Randers metrics which admit just two homogeneous geodesics were constructed, which shows that the present result is the best possible.

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