scholarly journals Stability of eϵ-Lipschitz and co-Lipschitz maps in Gromov–Hausdorff topology

2012 ◽  
Vol 231 (2) ◽  
pp. 774-797 ◽  
Author(s):  
Xiaochun Rong ◽  
Shicheng Xu
Keyword(s):  
Hausdorff Topology ◽  
Lipschitz Maps ◽  
Gromov Hausdorff Topology

Convergence of Riemannian 4-manifolds with L 2 L^{2} -curvature bounds

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Norman Zergänge
Keyword(s):  
Curvature Tensor ◽  
Curvature Flow ◽  
Riemannian Curvature ◽  
Hausdorff Topology ◽  
Curvature Bounds ◽  
Prove Theorem ◽  
Riemannian Curvature Tensor ◽  
Convergence Results ◽  
Uniform Diameter ◽  
Gromov Hausdorff Topology

Abstract In this work we prove convergence results of sequences of Riemannian 4-manifolds with almost vanishing {L^{2}} -norm of a curvature tensor and a non-collapsing bound on the volume of small balls. In Theorem 1.1 we consider a sequence of closed Riemannian 4-manifolds, whose {L^{2}} -norm of the Riemannian curvature tensor tends to zero. Under the assumption of a uniform non-collapsing bound and a uniform diameter bound, we prove that there exists a subsequence that converges with respect to the Gromov–Hausdorff topology to a flat manifold. In Theorem 1.2 we consider a sequence of closed Riemannian 4-manifolds, whose {L^{2}} -norm of the Riemannian curvature tensor is uniformly bounded from above, and whose {L^{2}} -norm of the traceless Ricci-tensor tends to zero. Here, under the assumption of a uniform non-collapsing bound, which is very close to the Euclidean situation, and a uniform diameter bound, we show that there exists a subsequence which converges in the Gromov–Hausdorff sense to an Einstein manifold. In order to prove Theorem 1.1 and Theorem 1.2, we use a smoothing technique, which is called {L^{2}} -curvature flow. This method was introduced by Jeffrey Streets. In particular, we use his “tubular averaging technique” in order to prove distance estimates of the {L^{2}} -curvature flow, which only depend on significant geometric bounds. This is the content of Theorem 1.3.

scholarly journals PROJECTIVE EMBEDDINGS AND LAGRANGIAN FIBRATIONS OF KUMMER VARIETIES

2009 ◽  
Vol 20 (05) ◽  
pp. 557-572
Author(s):  
YUICHI NOHARA
Keyword(s):  
Line Bundle ◽  
Ample Line Bundle ◽  
Theta Functions ◽  
Torus Action ◽  
Natural Basis ◽  
Tensor Power ◽  
Hausdorff Topology ◽  
Lagrangian Fibration ◽  
Holomorphic Sections ◽  
Gromov Hausdorff Topology

It is known that holomorphic sections of an ample line bundle L (and its tensor power Lk) over an Abelian variety A are given by theta functions. Moreover, a natural basis of the space of holomorphic sections of L or Lk is related to a certain Lagrangian fibration of A. In our previous paper, we studied projective embeddings of A defined by these basis for Lk. For a natural torus action on the ambient projective space, it is proved that its moment map, restricted to A, approximates the Lagrangian fibration of A for large k, with respect to the "Gromov–Hausdorff topology". In this paper, we prove that the same is true for the Kummer variety associated to A.

scholarly journals On dimensions of tangent cones in limit spaces with lower Ricci curvature bounds

2018 ◽  
Vol 2018 (742) ◽  
pp. 263-280
Author(s):  
Vitali Kapovitch ◽  
Nan Li
Keyword(s):  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Tangent Cones ◽  
Hölder Continuous ◽  
Hausdorff Topology ◽  
Curvature Bounds ◽  
Scale Measure ◽  
Gromov Hausdorff Topology ◽  
Bounded Below

Abstract We show that if X is a limit of n-dimensional Riemannian manifolds with Ricci curvature bounded below and γ is a limit geodesic in X, then along the interior of γ same scale measure metric tangent cones {T_{\gamma(t)}X} are Hölder continuous with respect to measured Gromov–Hausdorff topology and have the same dimension in the sense of Colding–Naber.

scholarly journals Metric topology on the moduli space

2021 ◽  
Vol 22 (1) ◽  
pp. 11
Author(s):  
Jialong Deng
Keyword(s):  
Moduli Space ◽  
Conformal Class ◽  
Hausdorff Topology ◽  
Metric Topology ◽  
Gromov Hausdorff Topology

<p>We define the smooth Lipschitz topology on the moduli space and show that each conformal class is dense in the moduli space endowed with Gromov-Hausdorff topology, which offers an answer to Tuschmann’s question.</p>

scholarly journals The structure of spaces with Bakry–Émery Ricci curvature bounded below

2019 ◽  
Vol 2019 (757) ◽  
pp. 1-50 ◽  
Author(s):  
Feng Wang ◽  
Xiaohua Zhu
Keyword(s):  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Ricci Solitons ◽  
Existence Problem ◽  
Fano Manifold ◽  
Hausdorff Topology ◽  
Limit Space ◽  
Gromov Hausdorff Topology ◽  
Bounded Below ◽  
Monge Ampere

AbstractWe explore the structure of limit spaces of sequences of Riemannian manifolds with Bakry–Émery Ricci curvature bounded below in the Gromov–Hausdorff topology. By extending the techniques established by Cheeger and Cloding for Riemannian manifolds with Ricci curvature bounded below, we prove that each tangent space at a point of the limit space is a metric cone. We also analyze the singular structure of the limit space as in a paper of Cheeger, Colding and Tian. Our results will be applied to study the limit spaces for a sequence of Kähler metrics arising from solutions of certain complex Monge–Ampère equations for the existence problem of Kähler–Ricci solitons on a Fano manifold via the continuity method.

scholarly journals On the gradient set of Lipschitz maps

2008 ◽  
Vol 2008 (625) ◽  
Author(s):  
Bernd Kirchheim ◽  
László Székelyhidi
Keyword(s):  
Lipschitz Maps

scholarly journals A DEFINABLE -ADIC ANALOGUE OF KIRSZBRAUN’S THEOREM ON EXTENSIONS OF LIPSCHITZ MAPS

2015 ◽  
Vol 17 (1) ◽  
pp. 39-57 ◽  
Author(s):  
Raf Cluckers ◽  
Florent Martin
Keyword(s):  
Direct Application ◽  
Lipschitz Maps ◽  
Lipschitz Map ◽  
Lipschitz Extensions ◽  
Topological Closure

A direct application of Zorn’s lemma gives that every Lipschitz map $f:X\subset \mathbb{Q}_{p}^{n}\rightarrow \mathbb{Q}_{p}^{\ell }$ has an extension to a Lipschitz map $\widetilde{f}:\mathbb{Q}_{p}^{n}\rightarrow \mathbb{Q}_{p}^{\ell }$. This is analogous to, but easier than, Kirszbraun’s theorem about the existence of Lipschitz extensions of Lipschitz maps $S\subset \mathbb{R}^{n}\rightarrow \mathbb{R}^{\ell }$. Recently, Fischer and Aschenbrenner obtained a definable version of Kirszbraun’s theorem. In this paper, we prove in the $p$-adic context that $\widetilde{f}$ can be taken definable when $f$ is definable, where definable means semi-algebraic or subanalytic (or some intermediary notion). We proceed by proving the existence of definable Lipschitz retractions of $\mathbb{Q}_{p}^{n}$ to the topological closure of $X$ when $X$ is definable.

Fuzzy Lipschitz maps and fixed point theorems in fuzzy metric spaces

2010 ◽  
Vol 161 (8) ◽  
pp. 1117-1130 ◽  
Author(s):  
Gabjin Yun ◽  
Seungsu Hwang ◽  
Jeongwook Chang
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Fuzzy Metric Spaces ◽  
Fuzzy Metric ◽  
Lipschitz Maps
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