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Daisuke Kazukawa

Publications and source records attributed to Daisuke Kazukawa.

10 recordsLinked to original sources
arXiv2024

Convergence of cones of metric measure spaces and its application to Cauchy distribution

We prove that the sequence of cones of metric measure spaces converges if the sequence of base spaces converges in Gromov's box, concentration, and weak topologies. As an application, we show that the generalized Cauchy distribution with suitable scaling converges to a half line in the concentration topology as the dimension diverges to infinity. This is a new example distinguished from previously known examples such as Gaussian distributions and typical closed Riemannian manifolds with constant Ricci curvature.

math.MG↗
arXiv2023

Invariants for Gromov's pyramids and their applications

Pyramids introduced by Gromov are generalized objects of metric spaces with Borel probability measures. We study non-trivial pyramids, where non-trivial means that they are not represented as metric measure spaces. In this paper, we establish general theory of invariants of pyramids and construct several invariants. Using them, we distinguish concrete pyramids. Furthermore, we study a space consisting of non-trivial pyramids and prove that the space have infinite dimension.

math.MG↗
arXiv2023

Topological aspects of the space of metric measure spaces

Gromov introduced two distance functions, the box distance and the observable distance, on the space of isomorphism classes of metric measure spaces and developed the convergence theory of metric measure spaces. We investigate several topological properties on the space equipped with these distance functions toward a deep understanding of convergence theory.

math.MG↗
arXiv2023

Principal bundle structure of the space of metric measure spaces

We study the topological structure of the space $\mathcal{X}$ of isomorphism classes of metric measure spaces equipped with the box or concentration topologies. We consider the scale-change action of the multiplicative group $\mathbb{R}_+$ of positive real numbers on $\mathcal{X}$, which has a one-point metric measure space, say $*$, as only one fixed-point. We prove that the $\mathbb{R}_+$-action on $\mathcal{X}_* := \mathcal{X} \setminus \{*\}$ admits the structure of nontrivial and locally trivial principal $\mathbb{R}_+$-bundle over the quotient space. Our bundle $\mathbb{R}_+ \to \mathcal{X}_* \to \mathcal{X}_*/\mathbb{R}_+$ is a curious example of a nontrivial principal fiber bundle with contractible fiber. A similar statement is obtained for the pyramidal compactification of $\mathcal{X}$, where we completely determine the structure of the fixed-point set of the $\mathbb{R}_+$-action on the compactification.

math.MG↗
arXiv2022

Boundedness of measured Gromov-Hausdorff precompact sets of metric measure spaces in pyramids

We prove that any measured Gromov-Hausdorff precompact set of metric measure spaces which is contained in a certain set, called a pyramid, is bounded by some metric measure space with respect to the Lipschitz order inside the pyramid. This is proved as a step towards a possible extension of the statement of Gromov, for which we gave a detailed proof in our previous work. Several related results are also obtained.

math.MG↗
arXiv2020

Convergence of metric transformed spaces

We consider the metric transformation of metric measure spaces/pyramids. We clarify the conditions to obtain the convergence of the sequence of transformed spaces from that of the original sequence, and, conversely, to obtain the convergence of the original sequence from that of the transformed sequence, respectively. As an application, we prove that spheres and projective spaces with standard Riemannian distance converge to a Gaussian space and the Hopf quotient of a Gaussian space, respectively, as the dimension diverges to infinity.

math.MG↗
arXiv2020

High-dimensional ellipsoids converge to Gaussian spaces

We prove the convergence of (solid) ellipsoids to a Gaussian space in Gromov's concentration/weak topology as the dimension diverges to infinity. This gives the first discovered example of an irreducible nontrivial convergent sequence in the concentration topology, where 'irreducible nontrivial' roughly means to be not constructed from Levy families nor box convergent sequences.

math.MG↗
arXiv2019

Concentration of product spaces

We investigate the relation between the concentration and the product of metric measure spaces. We have the natural question whether, for two concentrating sequences of metric measure spaces, the sequence of their product spaces also concentrates. A partial answer is mentioned in Gromov's book. We obtain a complete answer for this question.

math.MG↗
arXiv2018

Convergence of energy functionals and stability of lower bounds of Ricci curvature via metric measure foliation

The notion of the metric measure foliation is introduced by Galaz-García, Kell, Mondino, and Sosa. They studied the relation between a metric measure space with a metric measure foliation and its quotient space. They showed that the curvature-dimension condition and the Cheeger energy functional preserve from a such space to its quotient space. Via the metric measure foliation, we investigate the convergence theory for a sequence of metric measure spaces whose dimensions are unbounded.

math.MG↗