scholarly journals Symmetric rigidity for circle endomorphisms having bounded geometry

2021 ◽  
Author(s):  
John Adamski ◽  
Yunchun Hu ◽  
Yunping Jiang ◽  
Zhe Wang
Keyword(s):  
Bounded Geometry

scholarly journals Coarse median algebras: the intrinsic geometry of coarse median spaces and their intervals

2021 ◽  
Vol 27 (2) ◽  
Author(s):  
Graham A. Niblo ◽  
Nick Wright ◽  
Jiawen Zhang
Keyword(s):  
Finite Rank ◽  
Intrinsic Geometry ◽  
Bounded Geometry ◽  
Higher Dimensional ◽  
Median Algebra ◽  
Asymptotic Geometry ◽  
Median Operator

AbstractThis paper establishes a new combinatorial framework for the study of coarse median spaces, bridging the worlds of asymptotic geometry, algebra and combinatorics. We introduce a simple and entirely algebraic notion of coarse median algebra which simultaneously generalises the concepts of bounded geometry coarse median spaces and classical discrete median algebras. We study the coarse median universe from the perspective of intervals, with a particular focus on cardinality as a proxy for distance. In particular we prove that the metric on a quasi-geodesic coarse median space of bounded geometry can be constructed up to quasi-isometry using only the coarse median operator. Finally we develop a concept of rank for coarse median algebras in terms of the geometry of intervals and show that the notion of finite rank coarse median algebra provides a natural higher dimensional analogue of Gromov’s concept of $$\delta $$ δ -hyperbolicity.

scholarly journals WITTEN DEFORMATION FOR NONCOMPACT MANIFOLDS WITH BOUNDED GEOMETRY

2021 ◽  
pp. 1-38
Author(s):  
Xianzhe Dai ◽  
Junrong Yan
Keyword(s):  
Essential Role ◽  
Morse Function ◽  
Noncompact Manifold ◽  
Morse Inequalities ◽  
Bounded Geometry ◽  
Noncompact Manifolds ◽  
Witten Laplacian ◽  
Relative Cohomology ◽  
Witten Deformation ◽  
Small Eigenvalues

Abstract Motivated by the Landau–Ginzburg model, we study the Witten deformation on a noncompact manifold with bounded geometry, together with some tameness condition on the growth of the Morse function f near infinity. We prove that the cohomology of the Witten deformation $d_{Tf}$ acting on the complex of smooth $L^2$ forms is isomorphic to the cohomology of the Thom–Smale complex of f as well as the relative cohomology of a certain pair $(M, U)$ for sufficiently large T. We establish an Agmon estimate for eigenforms of the Witten Laplacian which plays an essential role in identifying these cohomologies via Witten’s instanton complex, defined in terms of eigenspaces of the Witten Laplacian for small eigenvalues. As an application, we obtain the strong Morse inequalities in this setting.

Bounded properties of curvatures of perturbed Ricci metric on curve moduli

2014 ◽  
Vol 25 (12) ◽  
pp. 1450113
Author(s):  
Xiaorui Zhu
Keyword(s):  
Finite Volume ◽  
Ricci Curvature ◽  
Moduli Space ◽  
Point Of View ◽  
Bisectional Curvature ◽  
Bounded Geometry ◽  
Holomorphic Bisectional Curvature ◽  
Chern Form ◽  
Thick Part ◽  
Kahler Metric

As is well-known, the Weil–Petersson metric ωWP on the moduli space ℳg has negative Ricci curvature. Hence, its negative first Chern form defines the so-called Ricci metric ωτ. Their combination [Formula: see text], C > 0, introduced by Liu–Sun–Yau, is called the perturbed Ricci metric. It is a complete Kähler metric with finite volume. Furthermore, it has bounded geometry. In this paper, we investigate the finiteness of this new metric from another point of view. More precisely, we will prove in the thick part of ℳg, the holomorphic bisectional curvature of [Formula: see text] is bounded by a constant depending only on the thick constant and C0 when C ≥ (3g - 3)C0, but not on the genus g.

scholarly journals Geodesic Random Walks, Diffusion Processes and Brownian Motion on Finsler Manifolds

2021 ◽  
Author(s):  
Tianyu Ma ◽  
Vladimir S. Matveev ◽  
Ilya Pavlyukevich
Keyword(s):  
Brownian Motion ◽  
Diffusion Process ◽  
Random Walks ◽  
Diffusion Processes ◽  
Riemannian Metric ◽  
Finsler Manifold ◽  
Finsler Manifolds ◽  
Bounded Geometry ◽  
And Brownian Motion

AbstractWe show that geodesic random walks on a complete Finsler manifold of bounded geometry converge to a diffusion process which is, up to a drift, the Brownian motion corresponding to a Riemannian metric.

scholarly journals POINCARÉ AND SOBOLEV INEQUALITIES FOR DIFFERENTIAL FORMS IN HEISENBERG GROUPS AND CONTACT MANIFOLDS

2020 ◽  
pp. 1-52
Author(s):  
ANNALISA BALDI ◽  
BRUNO FRANCHI ◽  
PIERRE PANSU
Keyword(s):  
Differential Form ◽  
Scale Invariance ◽  
Differential Forms ◽  
Sobolev Inequalities ◽  
Exact Form ◽  
Heisenberg Groups ◽  
Contact Manifolds ◽  
Bounded Geometry ◽  
Exterior Differential ◽  
Rumin Complex

Abstract In this paper, we prove contact Poincaré and Sobolev inequalities in Heisenberg groups $\mathbb{H}^{n}$ , where the word ‘contact’ is meant to stress that de Rham’s exterior differential is replaced by the exterior differential of the so-called Rumin complex $(E_{0}^{\bullet },d_{c})$ , which recovers the scale invariance under the group dilations associated with the stratification of the Lie algebra of $\mathbb{H}^{n}$ . In addition, we construct smoothing operators for differential forms on sub-Riemannian contact manifolds with bounded geometry, which act trivially on cohomology. For instance, this allows us to replace a closed form, up to adding a controlled exact form, with a much more regular differential form.

scholarly journals Coarse quotients by group actions and the maximal Roe algebra

2019 ◽  
Vol 11 (04) ◽  
pp. 875-907
Author(s):  
Logan Higginbotham ◽  
Thomas Weighill
Keyword(s):  
Automorphism Group ◽  
Metric Space ◽  
Large Scale ◽  
Countable Group ◽  
Crossed Product ◽  
Property A ◽  
Bounded Geometry ◽  
Roe Algebra ◽  
Scale Spaces ◽  
The Ideal

For a finitely generated group [Formula: see text] acting on a metric space [Formula: see text], Roe defined the warped space [Formula: see text], which one can view as a kind of large scale quotient of [Formula: see text] by the action of [Formula: see text]. In this paper, we generalize this notion to the setting of actions of arbitrary groups on large scale spaces. We then restrict our attention to what we call coarsely discontinuous actions by coarse equivalences and show that for such actions the group [Formula: see text] can be recovered as an appropriately defined automorphism group [Formula: see text] when [Formula: see text] satisfies a large scale connectedness condition. We show that for a coarsely discontinuous action of a countable group [Formula: see text] on a discrete bounded geometry metric space [Formula: see text] there is a relation between the maximal Roe algebras of [Formula: see text] and [Formula: see text], namely that there is a ∗-isomorphism [Formula: see text], where [Formula: see text] is the ideal of compact operators. If [Formula: see text] has Property A and [Formula: see text] is amenable, then [Formula: see text] has Property A, and thus the maximal Roe algebra and full crossed product can be replaced by the usual Roe algebra and reduced crossed product respectively in the above equation.

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