Latest Updates

* For the schedule of domestic events, see my Japanese page

Sep 1, 2025 Affiliation changed from Kyoto University to Fukuoka University

Aug 12, 2025 Updated a preprint "Alexandrov spaces are CS sets" on arXiv

Jul 21, 2025 Updated a preprint "Lipschitz homotopy convergence of Alexandrov spaces II" on arXiv (with Ayato Mitsuishi and Takao Yamaguchi)

Jul 11, 2025 Published a paper "A lower bound for the curvature integral under an upper curvature bound" in St. Petersburg Math. J. (English translation of Algebra i Analiz)

May 8, 2025 Updated a preprint "Topological regularity of Busemann spaces of nonpositive curvature" on arXiv (with Shijie Gu)

Research Interests

Keywords:  Sectional curvature / Alexandrov spaces / GCBA spaces / Gromov-Hausdorff convergence / Collapse / Extremal subsets / Busemann spaces

My main interests are in Riemannian manifolds with sectional curvature bounded below or above and their convergence theory. Under some more geometric constraints (e.g., diameter, injectivity radius, or volume bounds), these manifolds form a precompact family with respect to the Gromov-Hausdorff convergence, i.e., any sequence has a convergent subsequence. The general strategy of the convergence theory of Riemannian manifolds is to analyze their limit spaces by using curvature bounds to obtain feedback on the original manifolds.

The limits of Riemannian manifolds with sectional curvature bounded below are Alexandrov spaces (= CBB spaces) in the sense of Burago-Gromov-Perelman (1992). The limits of Riemannian manifolds with sectional curvature bounded above and injectivity radius bounded below are GCBA spaces (= geodesically complete CBA spaces) in the sense of Lytchak-Nagano (2019). These are metric spaces with sectional curvature bounded below or above, respectively, in a generalized sense. Typical examples are as follows:

• Alexandrov spaces — Euclidean cone with vertex angle not greater than 2π / boundary gluing of two or less sheets of half planes

• GCBA spaces — Euclidean cone with vertex angle not less than 2π / boundary gluing of two or more sheets of half planes

The above examples illustrate the dualities between the two geometries, but also reflect their differences: Alexandrov spaces are generally more restricted than GCBA spaces. I'm interested in the analogies and differences between lower and upper curvature bounds from the viewpoint of these limit spaces. For my results in this direction, see for instance Paper 10.

The biggest difference between the two convergence theories is the existence of collapse in the case of a lower curvature bound — the dimension may drop in the limit. To capture the collapsing phenomena, the notion of extremal subsets in Alexandrov spaces introduced by Perelman-Petrunin (1993) is essential. It is expected that a collapsing manifold admits a singular fibration structure over the limit space, where the singular fibers arise over the extremal subsets of the limit space. See for instance Paper 7.

Recently, I'm also interested in Busemann spaces, which are metric spaces with a weaker notion of nonpositive curvature. In the smooth setting, Busemann's curvature bound makes sense for Finsler manifolds, whereas Alexandrov-type curvature bounds only make sense for Riemannian manifolds. The difference between Alexandrov and Busemann curvature bounds exactly reflects the difference between Riemannian and Finsler geometries. See Paper 12.

Papers & Preprints

* In order of writing, not publishing  * The arXiv versions may differ slightly (but not substantially) from the published ones

Meaning of Tags:  R = Riemannian manifolds / A = Alexandrov spaces / G = GCBA spaces / C = Collapse / E = Extremal subsets / B = Busemann spaces

#12 Topological regularity of Busemann spaces of nonpositive curvature, with S. Gu, preprint (submitted), arXiv:2504.14455.  B G

#11 Alexandrov spaces are CS sets, preprint (submitted), arXiv:2404.14587.  A E

#10 A lower bound for the curvature integral under an upper curvature bound, Algebra i Analiz 36 (2024), no. 2, 131-160; St. Petersburg Math. J. 36 (2025), no. 2, 263-284; arXiv:2306.11577.  R G

#9 Lipschitz homotopy convergence of Alexandrov spaces II, with A. Mitsuishi and T. Yamaguchi, preprint (submitted), arXiv:2304.12515.  R A E

#8 Extremal subsets in geodesically complete spaces with curvature bounded above, Anal. Geom. Metr. Spaces 11 (2023), no. 1, 20230104; arXiv:2207.10369.  G E

#7 Euler characteristics of collapsing Alexandrov spaces, Arnold Math. J. 10 (2024), no. 4, 463-472; arXiv:2206.15104.  R A C E

#6 Collapsing to Alexandrov spaces with isolated mild singularities, Differential Geom. Appl. 85 (2022), 101951; arXiv:2108.11030.  R A C

#5 Noncritical maps on geodesically complete spaces with curvature bounded above, Ann. Global. Anal. Geom. 62 (2022), no. 3, 661-677; arXiv:2107.08859.  G

#4 Application of good coverings to collapsing Alexandrov spaces, Pacific J. Math. 316 (2022), no. 2, 335-365; arXiv:2010.02520.  R A C E

#3 A fibration theorem for collapsing sequences of Alexandrov spaces, J. Topol. Anal. 15 (2023), no. 1, 265-298; arXiv:1905.05484.  A C

#2 Regular points of extremal subsets in Alexandrov spaces, J. Math. Soc. Japan 74 (2022), no. 4, 1245-1268; arXiv:1905.05480.  A E

#1 Uniform boundedness on extremal subsets in Alexandrov spaces, J. Geom. Anal. 35 (2025), no. 2, 46, arXiv:1809.00603.  A C E

Summary

Paper 1 and Paper 2 studied the properties of extremal subsets in Alexandrov spaces. Paper 1 proved Gromov-type finiteness theorems for the number, Betti number, and volume of extremal subsets. Paper 2 developed the basic theory of regular points in extremal subsets. These are based on my master's thesis.

Paper 3 and Paper 4 extended two fibration theorems in CBB collapsing theory. Paper 3 generalized Yamaguchi's fibration theorem (1991) for collapsing Riemannian manifolds to a wide class of Alexandrov spaces. Paper 4 improved Perelman's fibration theorem (1997) in the case where the limit space has no proper extremal subsets. These are based on my doctoral thesis.

Paper 6 and Paper 7 are applications of the methods developed in Paper 3 and Paper 4, respectively. Paper 6 was an attempt to bridge the gap between the two fibration theorems mentioned above. Paper 7 proved a formula expressing the Euler characteristic of a collapsing manifold in terms of those of extremal subsets in the limit space and fibers over them. This solved a conjecture by Alesker (2018).

Paper 5 and Paper 8 explored GCBA counterparts of two important notions in CBB geometry. Paper 5 developed the GCBA version of the critical point theory for distance maps on Alexandrov spaces. Based on this, Paper 8 introduced the notion of an extremal subset in a GCBA space and exhibit some structural properties as well as its connection with topological singularities.

Paper 9, with Ayato Mitsuishi and Takao Yamaguchi, improved their Lipschitz homotopy stability theorem (2019) for noncollapsing Alexandrov spaces. Along the way, we obtained a Lipschitz version of Petersen's homotopy stability theorem (1990).

Paper 10 proved the CBA analogue of Petrunin's theorem (2008) on the uniform boundedness of the curvature integrals of CBB Riemannian manifolds. This is done by looking at the dual structure of the limit spaces, i.e., Alexandrov spaces and GCBA spaces.

Paper 11 showed that the stratification of an Alexandrov space by extremal subsets defines a CS structure in the sense of Siebenmann (1972). This refines the original work of Perelman-Petrunin and another result reveals further regularity of the stratification.

Paper 12, with Shijie Gu, extended the topological results of Lytchak-Nagano (2019, 2022) and Lytchak-Nagano-Stadler (2024) for CAT(0) spaces to Busemann spaces of nonpositive curvature. In particular, we proved that any Busemann 4-manifold is homeomorphic to Euclidean space, answering affirmatively a question of Gromov (1981).

Education & Employment

Sep 2025 -   Present Assistant Professor, Fukuoka University

Apr 2025 - Aug 2025 Program-Specific Assistant Professor, Kyoto University

Apr 2022 - Mar 2025 JSPS Postdoctoral Fellow, Osaka University (Host: Shin-ichi Ohta)

Apr 2022 - Sep 2022 Part-time Lecturer, Osaka University

Apr 2021 - Mar 2022 Part-time Postdoctoral Fellow and Teaching Assistant, Kyoto University (Host: Koji Fujiwara)

Apr 2017 - Mar 2021 Doctor of Science, Kyoto University (Advisor: Takao Yamaguchi)

Doctoral thesis "Fibration theorems for collapsing Alexandrov spaces," on which Paper 3 and Paper 4 are based

Apr 2017 - Mar 2018 Leave of absence

Apr 2015 - Mar 2017 Master of Science, Kyoto University (Advisor: Takao Yamaguchi)

Master's thesis "Several theorems on extremal subsets in Alexandrov spaces" (in Japanese), on which Paper 1 and Paper 2 are based

Apr 2011 - Mar 2015 Bachelor of Science, Kyoto University (Advisor: Takao Yamaguchi)

Grants

Jul 2025 - Mar 2027 JSPS KAKENHI #25K23336 "Geometry and convergence theory of upper and lower curvature bounds"

Apr 2022 - Mar 2025 JSPS KAKENHI #22KJ2099 (22J00100) "Geometry and collapsing theory of Alexandrov spaces"

Organization

Feb 9-20, 2026 Local organizer for The 18th MSJ-SI: Analysis, Geometry and Probability on Metric Measure Spaces

Sep 2025 -   Present Organizer for Fukuoka University Differential Geometry Seminar

Links

* For the list of talks, including abstracts and slides, see researchmap

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