scholarly journals A MATSUMOTO–MOSTOW RESULT FOR ZIMMER’S COCYCLES OF HYPERBOLIC LATTICES

2020 ◽  
Author(s):  
M. MORASCHINI ◽  
A. SAVINI
Keyword(s):  
Euler Number ◽  
Type Inequality ◽  
Surface Group ◽  
Closed Surface ◽  
Complete Characterization ◽  
Alternative Proof ◽  
Bounded Cohomology ◽  
Rigidity Result ◽  
Multiplicative Constant ◽  
Wood Type

AbstractFollowing the philosophy behind the theory of maximal representations, we introduce the volume of a Zimmer’s cocycle Γ × X → PO° (n, 1), where Γ is a torsion-free (non-)uniform lattice in PO° (n, 1), with n > 3, and X is a suitable standard Borel probability Γ-space. Our numerical invariant extends the volume of representations for (non-)uniform lattices to measurable cocycles and in the uniform setting it agrees with the generalized version of the Euler number of self-couplings. We prove that our volume of cocycles satisfies a Milnor–Wood type inequality in terms of the volume of the manifold Γ\ℍn. Additionally this invariant can be interpreted as a suitable multiplicative constant between bounded cohomology classes. This allows us to define a family of measurable cocycles with vanishing volume. The same interpretation enables us to characterize maximal cocycles for being cohomologous to the cocycle induced by the standard lattice embedding via a measurable map X → PO° (n, 1) with essentially constant sign.As a by-product of our rigidity result for the volume of cocycles, we give a different proof of the mapping degree theorem. This allows us to provide a complete characterization of maps homotopic to local isometries between closed hyperbolic manifolds in terms of maximal cocycles.In dimension n = 2, we introduce the notion of Euler number of measurable cocycles associated to a closed surface group and we show that it extends the classic Euler number of representations. Our Euler number agrees with the generalized version of the Euler number of self-couplings up to a multiplicative constant. Imitating the techniques developed in the case of the volume, we show a Milnor–Wood type inequality whose upper bound is given by the modulus of the Euler characteristic of the associated closed surface. This gives an alternative proof of the same result for the generalized version of the Euler number of self-couplings. Finally, using the interpretation of the Euler number as a multiplicative constant between bounded cohomology classes, we characterize maximal cocycles as those which are cohomologous to the one induced by a hyperbolization.

scholarly journals Chern Class Inequalities on Polarized Manifolds and Nef Vector Bundles

2020 ◽  
Author(s):  
Ping Li ◽  
Fangyang Zheng
Keyword(s):  
Chern Class ◽  
Vector Bundles ◽  
Euler Number ◽  
Type Inequality ◽  
Chern Number ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Bisectional Curvature ◽  
Chern Numbers ◽  
Nef Vector Bundles

Abstract This article is concerned with Chern class and Chern number inequalities on polarized manifolds and nef vector bundles. For a polarized pair $(M,L)$ with $L$ very ample, our 1st main result is a family of sharp Chern class inequalities. Among them the 1st one is a variant of a classical result and the equality case of the 2nd one is a characterization of hypersurfaces. The 2nd main result is a Chern number inequality on it, which includes a reverse Miyaoka–Yau-type inequality. The 3rd main result is that the Chern numbers of a nef vector bundle over a compact Kähler manifold are bounded below by the Euler number. As an application, we classify compact Kähler manifolds with nonnegative bisectional curvature whose Chern numbers are all positive. A conjecture related to the Euler number of compact Kähler manifolds with nonpositive bisectional curvature is proposed, which can be regarded as a complex analogue to the Hopf conjecture.

A Unified Version of Weighted Weak Type Inequality for Martingale Maximal Operators

2021 ◽  
Vol 58 (2) ◽  
pp. 216-229
Author(s):  
Yanbo Ren ◽  
Congbian Ma
Keyword(s):  
Weak Type ◽  
Type Inequality ◽  
Maximal Inequality ◽  
Complete Characterization ◽  
Maximal Operators ◽  
Weak Type Inequality

Let ɣ and Φ1 be nondecreasing and nonnegative functions defined on [0, ∞), and Φ2 is an N -function, u, v and w are weights. A unified version of weighted weak type inequality of the formfor martingale maximal operators f ∗ is considered, some necessary and su@cient conditions for it to hold are shown. In addition, we give a complete characterization of three-weight weak type maximal inequality of martingales. Our results generalize some known results on weighted theory of martingale maximal operators.

A Note on a Result of Ghosh and Sinha

1980 ◽  
Vol 29 (3-4) ◽  
pp. 169-172
Author(s):  
Bikas Kumar Sinha ◽  
Banshi Badan Mukhopadhyay
Keyword(s):  
Linear Model ◽  
Likelihood Ratio Test ◽  
Covariance Structure ◽  
Complete Characterization ◽  
Alternative Proof ◽  
Design Matrix ◽  
Ratio Test ◽  
Normal Linear Model ◽  
Insight Into

For the usual normal linear model with an lntraclass covariance structure, Ghosh and Sinha (1978) has given a complete characterization of tho design matrix for the robustness of the likelihood ratio test for linear hypotheses. We indicate here an alternative proof of the result which gives a better Insight into the problem.

Various Generalizations and Deformations of PSL(2,ℝ) Surface Group Representations and their Higgs Bundles

2018 ◽  
pp. 471-502
Author(s):  
Brian Collier
Keyword(s):  
Fundamental Group ◽  
Vector Bundles ◽  
Surface Group ◽  
Closed Surface ◽  
Connected Components ◽  
Character Variety ◽  
Higgs Bundles ◽  
Group Representations ◽  
Symmetric Powers ◽  
Higher Rank

The goal of this chapter is to examine the various ways in which Fuchsian representations of the fundamental group of a closed surface of genus g into PSL(2, R) and their associated Higgs bundles generalize to the higher-rank groups PSL(n, R), PSp(2n, R), SO0(2, n), SO0(n,n+1) and PU(n, n). For the SO0(n,n+1)-character variety, it parameterises n(2g−2) new connected components as the total spaces of vector bundles over appropriate symmetric powers of the surface, and shows how these components deform in the character variety. This generalizes results of Hitchin for PSL(2, R).

scholarly journals Variations on the theme of the uniform boundary condition

2019 ◽  
pp. 1-28 ◽  
Author(s):  
Daniel Fauser ◽  
Clara Löh
Keyword(s):  
Boundary Condition ◽  
Chain Complex ◽  
Alternative Proof ◽  
Cochain Complex ◽  
Bounded Cohomology ◽  
Simplicial Volume ◽  
Geometric Methods ◽  
Chain Level ◽  
Uniform Boundary Condition

The uniform boundary condition (UBC) in a normed chain complex asks for a uniform linear bound on fillings of null-homologous cycles. For the [Formula: see text]-norm on the singular chain complex, Matsumoto and Morita established a characterization of the UBC in terms of bounded cohomology. In particular, spaces with amenable fundamental group satisfy the UBC in every degree. We will give an alternative proof of statements of this type, using geometric Følner arguments on the chain level instead of passing to the dual cochain complex. These geometric methods have the advantage that they also lead to integral refinements. In particular, we obtain applications in the context of integral foliated simplicial volume.

scholarly journals An Alternative Proof of Rubin's Lemma

2020 ◽  
Vol 1 ◽  
pp. 51-54
Author(s):  
Santosh Ghimire
Keyword(s):  
Probabilistic Approach ◽  
Type Inequality ◽  
Alternative Proof ◽  
Theoretic Approach

Rubin's Lemma is an inhomogenous type inequality which is satisfied by the sequence of dyadic martingales. In this paper, we give a proof using the measure theoretic approach which is simpler and different than the original probabilistic approach.

On geodesically reversible Finsler manifolds

2021 ◽  
pp. 1-17
Author(s):  
Yong Fang
Keyword(s):  
Positive Curvature ◽  
Closed Surface ◽  
Natural Generalization ◽  
Finsler Metric ◽  
Finsler Metrics ◽  
Closed Formula ◽  
Rigidity Result ◽  
Projectively Flat ◽  
Constant Positive Curvature ◽  
Randers Metrics

A Finsler manifold is said to be geodesically reversible if the reversed curve of any geodesic remains a geometrical geodesic. Well-known examples of geodesically reversible Finsler metrics are Randers metrics with closed [Formula: see text]-forms. Another family of well-known examples are projectively flat Finsler metrics on the [Formula: see text]-sphere that have constant positive curvature. In this paper, we prove some geometrical and dynamical characterizations of geodesically reversible Finsler metrics, and we prove several rigidity results about a family of the so-called Randers-type Finsler metrics. One of our results is as follows: let [Formula: see text] be a Riemannian–Finsler metric on a closed surface [Formula: see text], and [Formula: see text] be a small antisymmetric potential on [Formula: see text] that is a natural generalization of [Formula: see text]-form (see Sec. 1). If the Randers-type Finsler metric [Formula: see text] is geodesically reversible, and the geodesic flow of [Formula: see text] is topologically transitive, then we prove that [Formula: see text] must be a closed [Formula: see text]-form. We also prove that this rigidity result is not true for the family of projectively flat Finsler metrics on the [Formula: see text]-sphere of constant positive curvature.

scholarly journals Geodesic stretch, pressure metric and marked length spectrum rigidity

2021 ◽  
pp. 1-49
Author(s):  
COLIN GUILLARMOU ◽  
GERHARD KNIEPER ◽  
THIBAULT LEFEUVRE
Keyword(s):  
Limit Theorem ◽  
Uniform Estimate ◽  
Length Spectrum ◽  
Alternative Proof ◽  
Geodesic Flows ◽  
Rigidity Result ◽  
Local Rigidity ◽  
The Central Limit Theorem ◽  
Marked Length Spectrum ◽  
Anosov Flows

Abstract We refine the recent local rigidity result for the marked length spectrum obtained by the first and third author in [GL19] and give an alternative proof using the geodesic stretch between two Anosov flows and some uniform estimate on the variance appearing in the central limit theorem for Anosov geodesic flows. In turn, we also introduce a new pressure metric on the space of isometry classes, which reduces to the Weil–Petersson metric in the case of Teichmüller space and is related to the works [BCLS15, MM08].

scholarly journals Non-injective representations of a closed surface group into PSL(2, )

2007 ◽  
Vol 142 (2) ◽  
pp. 289-304 ◽  
Author(s):  
LOUIS FUNAR ◽  
MAXIME WOLFF
Keyword(s):  
Fundamental Group ◽  
Euler Class ◽  
Discrete Subgroup ◽  
Surface Group ◽  
Closed Surface ◽  
The Other ◽  
Connected Components ◽  
Discrete Representation ◽  
Discrete Representations

AbstractLet e denote the Euler class on the space ${\text{\rm Hom}(\Gamma_g,{PSL(2,{\mathbb R})}}$ of representations of the fundamental group Γg of the closed surface Σg of genus g. Goldman showed that the connected components of ${\text{\rm Hom}(\Gamma_g,{PSL(2,{\mathbb R})}}$ are precisely the inverse images e−1(k), for 2−2g≤ k≤ 2g−2, and that the components of Euler class 2−2g and 2g−2 consist of the injective representations whose image is a discrete subgroup of ${PSL(2,{\mathbb R})}$. We prove that non-faithful representations are dense in all the other components. We show that the image of a discrete representation essentially determines its Euler class. Moreover, we show that for every genus and possible corresponding Euler class, there exist discrete representations.

scholarly journals On a new generalization of some Hilbert-type inequalities

2021 ◽  
Vol 19 (1) ◽  
pp. 569-582
Author(s):  
Minghui You ◽  
Wei Song ◽  
Xiaoyu Wang
Keyword(s):  
Kernel Function ◽  
Euler Number ◽  
Bernoulli Number ◽  
Type Inequality ◽  
Trigonometric Functions ◽  
Hardy Type Inequality ◽  
Hardy Type ◽  
Higher Derivatives ◽  
Derivatives Of ◽  
Hilbert Type

Abstract In this work, by introducing several parameters, a new kernel function including both the homogeneous and non-homogeneous cases is constructed, and a Hilbert-type inequality related to the newly constructed kernel function is established. By convention, the equivalent Hardy-type inequality is also considered. Furthermore, by introducing the partial fraction expansions of trigonometric functions, some special and interesting Hilbert-type inequalities with the constant factors represented by the higher derivatives of trigonometric functions, the Euler number and the Bernoulli number are presented at the end of the paper.

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