Morse quasiflats I

Huang, Jingyin; Kleiner, Bruce; Stadler, Stephan

doi:10.1515/crelle-2021-0073

Cited by 9 publications

(3 citation statements)

References 98 publications

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“…The (non-relative) asymptotic Plateau problem has recently been solved by Kleiner-Lang [KL20] and we use their work extensively throughout this section. The idea to use solutions to a relative Plateau problem originated from [HKS22].…”

Section: Rigidity In Rankmentioning

confidence: 99%

CAT(0) spaces of higher rank

Stadler¹

2022

Preprint

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Ballmann's Rank Rigidity Conjecture predicts that a CAT(0) space of higher rank with a geometric group action is rigid -isometric to a Riemannian symmetric space, a Euclidean building, or splits as a direct product. We confirm this conjecture in rank 2. For CAT(0) spaces of higher rank n ≥ 2 we prove rigidity if the space contains a periodic n-flat and every complete geodesic lies in some n-flat. This does not require a geometric group action.

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Section: Rigidity In Rankmentioning

confidence: 99%

CAT(0) spaces of higher rank

Stadler¹

2022

Preprint

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“…Metric currents. References for this section are [AK + 00], [HKS22] and [Wen05]. Ambrosio and Kirchheim extended the classical theory of normal and integral currents developped by Federer and Fleming [FF60] to arbitrarily complete metric spaces.…”

Section: Introductionmentioning

confidence: 99%

Coarse embeddings of symmetric spaces and Euclidean buildings

Bensaid¹

2022

Preprint

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Introduced by Gromov in the 80's, coarse embeddings are a generalization of quasi-isometric embeddings when the control functions are not necessarily affine. In this paper, we will be particularly interested in coarse embeddings between symmetric spaces and Euclidean buildings. The quasi-isometric case is very well understood thanks to the rigidity results for symmetric spaces and buildings of higher rank by Anderson-Schroeder, Kleiner, Kleiner-Leeb and Eskin-Farb in the 90's. In particular, it is well known that the rank of these spaces is monotonous under quasi-isometric embeddings. This is no longer the case for coarse embeddings as shown by horospherical embeddings. However, we show that in the absence of a Euclidean factor in the domain, the rank is monotonous under coarse embeddings. This answers a question by David Fisher and Kevin Whyte. We can also relax the condition on the domain by allowing it to contain a Euclidean factor of dimension 1, answering a question by Gromov. OUSSAMA BENSAID 4.4. Proof of Theorem 1.10 38 5. When the domain X has a one-dimensional Euclidean factor 49 5.1. Coarse embeddings of Euclidean spaces into lower rank 49 6. Further questions 54 References 55

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“…More recently, Ambrosio-Kirchheim [2] generalized Federer-Fleming's theory to the setting of metric spaces, resulting in a rich and powerful theory of metric integral currents in complete metric spaces. We refer to the articles [13,26,32,46,48,53] for some recent applications of this theory in various settings. In the present article, we show that a metric n-manifold satisfying weak assumptions such as linear local contractibility admits an essentially unique integral n-current without boundary.…”

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confidence: 99%

Geometric and analytic structures on metric spaces homeomorphic to a manifold

Basso¹,

Marti²,

Wenger³

2023

Preprint

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We study metric spaces homeomorphic to a closed, oriented manifold from both geometric and analytic perspectives. We show that such spaces (which are sometimes called metric manifolds) admit a non-trivial integral current without boundary, provided they satisfy some weak assumptions. The existence of such an object should be thought of as an analytic analog of the fundamental class of the space and can also be interpreted as giving a way to make sense of Stokes' theorem in this setting. We use this to establish (relative) isoperimetric inequalities in metric n-manifolds that are Ahlfors n-regular and linearly locally contractible. As an application, we obtain a short and conceptually simple proof of a deep theorem of Semmes about the validity of Poincaré inequalities in these spaces. In the smooth case the idea for such a proof goes back to Gromov. We also give sufficient conditions for a metric manifold to be rectifiable.

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Morse quasiflats I

Cited by 9 publications

References 98 publications

CAT(0) spaces of higher rank

CAT(0) spaces of higher rank

Coarse embeddings of symmetric spaces and Euclidean buildings

Geometric and analytic structures on metric spaces homeomorphic to a manifold

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