A generalized tetrahedral property
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Mathematische Zeitschrift
A generalized tetrahedral property Jesús Núñez-Zimbrón1,2 · Raquel Perales3 Received: 23 December 2018 / Accepted: 26 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We present examples of metric spaces that are not Riemannian manifolds nor dimensionally homogeneous that satisfy Sormani’s Tetrahedral Property. We then note that Euclidean cones over metric spaces with small diameter do not satisfy this property. Therefore, we extend the tetrahedral property to a less restrictive one and prove that this generalized definition retains all the results of the original tetrahedral property proven by Portegies–Sormani: it provides a lower bound on the sliced filling volume and a lower bound on the volumes of balls. Thus, sequences with uniform bounds on this Generalized Tetrahedral Property also have subsequences which converge in both the Gromov–Hausdorff and Sormani–Wenger intrinsic flat sense to the same noncollapsed and countably rectifiable limit space.
1 Introduction 1.1 Historical context The n-dimensional (C, β)-tetrahedral property for a metric space was originally defined in [10]. As this property is rather strong an integral version was also introduced. These properties are given in terms of distances between points in metric spheres and provide a lower bound on the volumes of balls. Thus, a Gromov–Hausdorff (Integral) Tetrahedral Compactness Theorem for sequences of n-dimensional Riemannian manifolds with a uniform upper bound on volume and diameter that satisfy a uniform (integral) tetrahedral property was deduced. Furthermore, it is shown that the limits are countably Hn -rectifiable metric spaces [9], where Hn denotes the n-dimensional Hausdorff measure. The proof of the (Integral) Tetrahedral Compactness Theorems is based upon intrinsic flat convergence. The intrinsic flat distance between integral current spaces, countably Hn -
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Raquel Perales [email protected] Jesús Núñez-Zimbrón [email protected]
1
Department of Mathematics, University of California, Santa Barbara, CA 93106, USA
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Present Address: Centro de Investigación en Matemáticas, Jalisco S/N, 36023 Guanajuato, Guanajuato, Mexico
3
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Oaxaca, Mexico
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J. Núñez-Zimbrón, R. Perales
rectifiable metric spaces that are generalizations of oriented manifolds, was introduced by Sormani–Wenger in [12]. It was defined using Gromov’s idea of isometrically embedding two metric spaces into a common metric space. However, rather than measuring the Hausdorff distance between the images, since they are integral currents in the sense of Ambrosio– Kirchheim [1] one takes the flat distance between them. A compactness theorem with respect to the intrinsic flat distance for the class of n-dimensional integral current spaces holds [14]. Thus, limits obtained with this distance are automatically either countably Hn -rectifiable or the n-dimensional zero integral current space. Moreover, once a sequence of integral c
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