scholarly journals Two versions of the uniform Roe algebras for spaces without bounded geometry

Author(s):  
V. Manuilov ◽  
J. Zhu
2021 ◽  
Vol 27 (2) ◽  
Author(s):  
Graham A. Niblo ◽  
Nick Wright ◽  
Jiawen Zhang

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.


Author(s):  
Xianzhe Dai ◽  
Junrong Yan

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.


2018 ◽  
Vol 459 (2) ◽  
pp. 686-716 ◽  
Author(s):  
Pere Ara ◽  
Kang Li ◽  
Fernando Lledó ◽  
Jianchao Wu
Keyword(s):  

2014 ◽  
Vol 25 (12) ◽  
pp. 1450113
Author(s):  
Xiaorui Zhu

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.


Author(s):  
Tianyu Ma ◽  
Vladimir S. Matveev ◽  
Ilya Pavlyukevich

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.


Author(s):  
ANNALISA BALDI ◽  
BRUNO FRANCHI ◽  
PIERRE PANSU

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.


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