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Koichi Nagano

Publications and source records attributed to Koichi Nagano.

11 recordsLinked to original sources
arXiv2026

Asymptotic geometric regularity of CAT(0) spaces

We prove that if an n-dimensional geodesically complete CAT(0) space has Tits boundary sufficiently close to the (n-1)-dimensional standard unit sphere, then it is bi-Lipschiz homeomorphic to the n-dimensional Euclidean space. As an application, we conclude that if an (n-1)-dimensional geodesically complete CAT(1) space is sufficiently close to the (n-1)-dimensional standard unit sphere, then they are bi-Lipschiz homeomorphic to each other.

math.DG↗
arXiv2026

Wall singularity of spaces with an upper curvature bound

We study typical wall singularity of codimension one for locally compact geodesically complete metric spaces with an upper curvature bound. We provide a geometric structure theorem of codimension one singularity, and a geometric characterization of codimension two regularity. These give us necessary and sufficient conditions for singular sets to be of codimension at least two.

math.DG↗
arXiv2023

Two-dimensional metric spaces with curvature bounded above II

As a continuation of \cite{NSY:local}, we mainly discuss the global structure of two-dimensional locally compact geodesically complete metric spaces with curvature bounded above. We first obtain the result on the Lipschitz homotopy approximations of such spaces by polyhedral spaces. We define the curvature measures on our spaces making use of the convergence of the curvature measures, and establish Gauss-Bonnet Theorem. We also give a characterization of such spaces.

math.MG↗
arXiv2023

Two-dimensional metric spaces with curvature bounded above I

We determine the local geometric structure of two-dimensional metric spaces with curvature bounded above as the union of finitely many properly embedded/branched immersed Lipschitz disks. As a result, we obtain a graph structure of the topological singular point set of such a singular surface.

math.MG↗
arXiv2021

Asymptotic topological regularity of CAT(0) spaces

We study asymptotic topological regularity of CAT(0) spaces. We prove that if a purely n-dimensional, proper, geodesically complete CAT(0) space has small volume growth, then it is homeomorphic to the n-dimensional Euclidean space. We also discuss asymptotic geometry of proper, geodesically complete CAT(0) spaces of small volume growth.

math.DG↗
arXiv2021

Volume pinching theorems for CAT(1) spaces

We examine volume pinching problems of CAT(1) spaces. We characterize a class of compact geodesically complete CAT(1) spaces of small specific volume. We prove a sphere theorem for compact CAT(1) homology manifolds of small volume. We also formulate a criterion of manifold recognition for homology manifolds on volume growths under an upper curvature bound.

math.DG↗
arXiv2018

Topological regularity of spaces with an upper curvature bound

We prove that a locally compact space with an upper curvature bound is a topological manifold if and only if all of its spaces of directions are homotopy equivalent and not contractible. We discuss applications to homology manifolds, limits of Riemannian manifolds and deduce a sphere theorem.

math.DG↗
arXiv2018

Geodesically complete spaces with an upper curvature bound

We study geometric and topological properties of locally compact, geodesically complete spaces with an upper curvature bound. We control the size of singular subsets, discuss homotopical and measure-theoretic stratifications and regularity of the metric structure on a large part.

math.DG↗
arXiv2004

Fixed point sets of parabolic isometries of CAT(0)-spaces

We study the fixed point set in the ideal boundary of a parabolic isometry of a proper CAT(0)-space. We show that the radius of the fixed point set is at most pi/2, and study its centers. As a consequence, we prove that the set of fixed points is contractible with respect to the Tits topology.

math.DG↗