Papers by Daniel Luckhardt
We show that assuming lower bounds on the Ricci curvature and the injectivity radius the absolute... more We show that assuming lower bounds on the Ricci curvature and the injectivity radius the absolute value of any characteristic number of a Riemannian manifold M is bounded proportionally to the volume. The proof relies on the definition of a connection in terms of an harmonic Hölder regular metric tensor and a cover. We also remark on a volume comparison theorem for Betti numbers as a consequence of a result by Bowen.
We investigate regularization of riemannian metrics by mollification. Assuming both-sided bounds ... more We investigate regularization of riemannian metrics by mollification. Assuming both-sided bounds on the Ricci tensor and a lower injectivity radius bound we obtain a uniform estimate on the change of the sectional curvature. Actually, our result holds for any metric with a uniform bound on the W^2,p-harmonic radius.
We generalize the concept of a norm on a vector space to one of a norm on a category. This provid... more We generalize the concept of a norm on a vector space to one of a norm on a category. This provides a unified perspective on many specific matters in many different areas of mathematics like set theory, functional analysis, measure theory, topology, and metric space theory. We will especially address the two last areas in which the monotone-light factorization and, respectively, the Gromov-Hausdorff distance will naturally appear. In our formalization a Schr\"oder-Bernstein property becomes an axiom of a norm which constitutes interesting properties of the categories in question. The proposed concept provides a convenient framework for metrizations.
arXiv: Differential Geometry, 2020
We investigate regularization of riemannian metrics by mollification. Assuming both-sided bounds ... more We investigate regularization of riemannian metrics by mollification. Assuming both-sided bounds on the Ricci tensor and a lower injectivity radius bound we obtain a uniform estimate on the change of the sectional curvature. Actually, our result holds for any metric with a uniform bound on the $W^{2,p}$-harmonic radius.
We study a weak form of Gromov-Hausdorff convergence of Riemannian manifolds, also known as Benja... more We study a weak form of Gromov-Hausdorff convergence of Riemannian manifolds, also known as Benjamini-Schramm convergence. This concept is also applicable to other areas and has widely been studied in the context of graphs. The main result is the continuity of characteristic numbers normalized by the volume with respect to the Benjamini-Schramm topology on the class of Riemannian manifolds with a uniform lower bound on injectivity radius and Ricci curvature. An immediate consequence is a comparison theorem that gives for any characteristic number a linear bound in terms of the volume on the entire class of manifolds mentioned. We give another interpretation of the result showing that characteristic numbers can be reconstructed with some accuracy from local random information.
We show that assuming lower bounds on the Ricci curvature and the injectivity radius the absolute... more We show that assuming lower bounds on the Ricci curvature and the injectivity radius the absolute value of any characteristic number of a Riemannian manifold $M$ is bounded proportionally to the volume. The proof relies on the definition of a connection in terms of an harmonic Holder regular metric tensor and a cover. We also remark on a volume comparison theorem for Betti numbers as a consequence of a result by Bowen.
We generalize the concept of a norm on a vector space to one of a norm on a category. This provid... more We generalize the concept of a norm on a vector space to one of a norm on a category. This provides a unified perspective on many specific matters in many different areas of mathematics like set theory, functional analysis, measure theory, topology, and metric space theory. We will especially address the two last areas in which the monotone-light factorization and, respectively, the Gromov-Hausdorff distance will naturally appear. In our formalization a Schröder-Bernstein property becomes an axiom of a norm which constitutes interesting properties of the categories in question. The proposed concept provides a convenient framework for metrizations.
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Papers by Daniel Luckhardt