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Ayato Mitsuishi

Publications and source records attributed to Ayato Mitsuishi.

16 recordsLinked to original sources
arXiv2025

Lipschitz homotopy convergence of Alexandrov spaces II

We establish a quantitative version of the Lipschitz homotopy convergence introduced by Mitsuishi and Yamaguchi for a moduli space of compact Alexandrov spaces without collapsing. Along the way, we obtain a Lipschitz version of Petersen's homotopy stability theorem that is applicable to more general settings, including CAT spaces. We also show that the Lipschitz homotopies can be chosen to preserve the singular strata of Alexandrov spaces, i.e., extremal subsets.

math.DG↗
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↗
arXiv2022

Principal eigenvalue problem for infinity Laplacian in metric spaces

This paper is concerned with the Dirichlet eigenvalue problem associated to the $\infty$-Laplacian in metric spaces. We establish a direct PDE approach to find the principal eigenvalue and eigenfunctions in a proper geodesic space without assuming any measure structure. We provide an appropriate notion of solutions to the $\infty$-eigenvalue problem and show the existence of solutions by adapting Perron's method. Our method is different from the standard limit process via the variational eigenvalue formulation for $p$-Laplacian in the Euclidean space.

math.AP↗
arXiv2020

Distance functions on convex bodies and symplectic toric manifolds

In this paper we discuss three distance functions on the set of convex bodies. In particular we study the convergence of Delzant polytopes, which are fundamental objects in symplectic toric geometry. By using these observations, we derive some convergence theorems for symplectic toric manifolds with respect to the Gromov-Hausdorff distance.

math.MG↗
arXiv2019

Certain min-max values related to the $p$-energy and packing radii of Riemannian manifolds and metric measure spaces

Grosjean proved that the $(1/p)$-th power of the first eigenvalue of the $p$-Laplacian on a closed Riemannian manifold converges to the twice of the inverse of the diameter of the space, as $p \to \infty$. Before this, a corresponding result for the Dirichlet first eigenvalues was also obtained by Juutinen, Lindqvist and Manfredi. We extend those results for certain $k$-th min-max value related to the $p$-energy, where the corresponding limits are packing radii introduced by Grove-Markvorsen or its variant. Furthermore, we remark that our result holds for more singular setting.

math.DG↗
arXiv2018

Good coverings of Alexandrov spaces

In the present paper, we define a notion of good coverings of Alexandrov spaces with curvature bounded below, and prove that every Alexandrov space admits such a good covering and that it has the same homotopy type as the nerve of the good covering. We also prove the stability of the isomorphism classes of the nerves of good coverings in the non-collapsing case. In the proof, we need a version of Perelman's fibration theorem, which is also proved in this paper.

math.MG↗
arXiv2018

Lipschitz homotopy convergence of Alexandrov spaces

We introduce the notion of good coverings of metric spaces, and prove that if a metric space admits a good covering, then it has the same locally Lipschitz homotopy type as the nerve complex of the covering. As an application, we obtain a Lipschitz homotopy stability result for a moduli space of compact Alexandrov spaces without collapsing.

math.MG↗
arXiv2018

Obtuse constants of Alexandrov spaces

We introduce a new geometric invariant called the obtuse constant of spaces with curvature bounded below. We first find relations between this invariant and the normalized volume. We also discuss the case of maximal obtuse constant equal to $π/2$, where we prove some rigidity for spaces. Although we consider Alexandrov spaces with curvature bounded below, the results are new even in the Riemannian case.

math.DG↗
arXiv2016

Orientability and fundamental classes of Alexandrov spaces with applications

In the present paper, we consider several valid notions of orientability of Alexandov spaces and prove that all such conditions are equivalent. Further, we give topological and geometric applications of the orientability. In particular, a Poincaré-type duality theorem is proved. As a corollary to the duality theorem, we also prove that if a closed Alexandrov space admits a positive curvature bound in a synthetic sense, then its codimension one homology vanishes. Further, we obtain a filling radius inequality for closed orientable Alexandrov spaces.

math.MG↗
arXiv2016

Self and partial gluing theorems for Alexandrov spaces with a lower curvature bound

This paper is devoted to prove that if an Alexandrov space of curvature not less than $κ$ with a codimension one extremal subset which admits an isometric involution with respect to the induced length metric, then the metric space obtained by gluing the extremal subset along the isometry is an Alexandrov space of curvature not less than $κ$. This is a generalization of Perelman's doubling and Petrunin's gluing theorems.

math.MG↗
arXiv2015

The coincidence of the current homology and the measure homology via a new topology on spaces of Lipschitz maps

We consider the category of all locally Lipschitz contractible metric spaces and all locally Lipschitz maps, which is a wide class of metric spaces, including all finite dimensional Alexandrov spaces and all CAT spaces. We also consider the chain complex of normal currents with compact support in a metric space in the sense of Ambrosio and Kirchheim. In the present paper, its homology is proved to be a homotopy invariant on the category. To prove this result, we define a new topology on a space of Lipschitz maps between arbitrary metric spaces. This topology is proved to coincide with the usual $C^1$-topology on the space of $C^1$-maps between compact Riemannian manifolds.

math.AT↗
arXiv2015

The coincidence of the homologies of integral currents and of integral singular chains, via cosheaves

We consider the notion of metric spaces being locally Lipschitz contractible introduced by Yamaguchi, and a category of metric spaces satisfying this condition. Many objects in metric geometry including CAT-spaces and Alexandrov spaces, belong to this category. We consider the homology of integral currents with compact support in a metric space, introduced by Ambrosio and Kirchheim, and prove that it and the usual integral singular homology are isomorphic on the category. The proof of it is based on the theory of cosheaves. A method to compare the homologies associated to cosheaves is also proved in this paper.

math.AT↗