FY2021 Annual Report
Analysis on Metric Spaces Unit
Assistant Professor Xiaodan Zhou
Abstract
Our unit continues the study of geometric function theory and partial differential equations on subRiemannian manifold and general metric measure spaces together with our collaborators in the past year. In particular, we obtain results helping us detect the Sobolev regulairty of a mapping by checking its infinitesimal distortion outside a small set which provides theorems even new in the classical Euclidean spaces. In the setting of Heisenberg group, we find a way to construct quasiconvex enevelope of a given function by iteratively solving a PDE. This helps us find the convex hull of a set and also leads to other applications. We also show a gamebased interpretation of HamiltonJacobiIsaacs equations and study asymptotically mean value harmonic functions in metric spaces. 6 paper have been submitted in the past year and 8 talks (7 invited and 1 contributed talk) are delivered in seminars and conferences around the world.
The past year is also a productive year for our unit in expanding both research and outreach collaboration. We launched our unit seminar from May, 2021 and hosted 12 seminar talks in the past fiscal year, including 2 onsite talks and 10 online talks. Two mathematicians are inivetd to OIST in March 2022 to give minicourses on optimal transport and we also made the courses accessible both to the OIST community and online participants. Another highlight is that together with 4 other organizers from Tokyo, Kyoto and Hiroshima, we launched the first online monthly math colloquium in Japan which is devoted to offering broad overview mathematical talks for a general audience and a panel discussion session focusing on various topics concerning diversity, equity and inclusion issues.
Two grants are awarded to our unit in the past year including a JSPS GrantinAid for EarlyCareer Scientists (April 2022March 2025) and Rita R. Colwell Impact Fund from OIST for organizing Women at the intersection of mathematics and theoretical physics meet in Okinawain in September, 2022.
Two graduate students joined our unit as rotation students during January to April, 2022. We look forward to welcoming more postdocs and students to join in the upcoming years.
1. Staff
 Dr. Antoni Kijowski, Postdoctoral Scholar
 Chiyo Eto, Administrative Assistant
Alumni
 Rotation Graduate stucent, Geoffrey Garcia JanuaryApril, 2022
 Rotation Graduate student, Jonas Schneider JanuaryApril, 2022
2. Collaborations
2.1 Collaborators
 Panu Lahti, Chinese Academy of Sciences
 Projects:
 Quasiconformal and Sobolev mappings in nonAhlfors regular metric spaces
 Absolutely continuous mappings on doubling metric measure spaces
 Qing Liu, Fukuoka University
 Projects:
 Differential games and HamiltonJacobiIsaacs equations in metric spaces
 Hoorizontally quasiconvex envelope in the Heisenberg group
 Tomasz Adamowicz, Institute of Mathematics of the Polish Academy of Sciences and Elefterios Soultanis, Radboud University
 Projects:
 Asymptotically mean value harmonic functions in doubling metric measure spaces, subriemannian and RCD settings
3. Activities and Findings
3.1 Quasiconformal and Sobolev mappings in nonAhlfors regular metric spaces
We show that a mapping f : X → Y satisfying the metric condition of quasiconformality outside suitable exceptional sets is in the NewtonSobolev class \(f\in N^{1,1}_{loc}(X;Y)\). Contrary to previous works, we only assume an asymptotic version of Ahlforsregularity on X, Y . This allows many nonAhlfors regular spaces, such as weighted spaces and Fred Gehring’s bowtie, to be included in the theory. Unexpectedly, already in the classical setting of unweighted Euclidean spaces, our theory detects Sobolev mappings that are not recognized by previous results.
Given a homeomorphism f : X → Y between Qdimensional spaces X, Y , we show that f satisfying the metric definition of quasiconformality outside suitable exceptional sets implies that \(f\in N^{1,p}_{loc}(X;V)\), where 1 < p ≤ Q, and also implies one direction of the geometric definition of quasiconformality. Unlike previous results, we only assume a pointwise version of Ahlfors Qregularity, which in particular enables various weighted spaces to be included in the theory. Unexpectedly, we can apply this to prove results that are new even in the classical Euclidean setting. In particular, in many cases we are able to prove \(f\in N^{1,Q}_{loc}(X;V)\) without the strong assumption \(h_f\in L^\infty(X)\).
3.2 Absolutely continuous mappings on doubling metric measure spaces
Following Maly’s definition of absolutely continuous functions of several variables, we consider Qabsolutely continuous mappings f : X → V between a doubling metric measure space X and a Banach space V . The relation between these mappings and Sobolev mappings \(f\in N^{1,p}_{loc}(X;V)\) for p≥Q is investigated. In particular, a locally Qabsolutely continuous mapping on an Ahlfors Qregular space is a continuous mapping in\(N^{1,p}_{loc}(X;V)\)), as well as differentiable almost everywhere in terms of Cheeger derivatives provided V satisfies the RadonNikodym property. Conversely, though a continuous Sobolev mapping\(N^{1,p}_{loc}(X;V)\)) is generally not locally Qabsolutely continuous, this implication holds if f is further assumed to be pseudomonotone. It follows that pseudomonotone mappings satisfying a relaxed quasiconformality condition are also Qabsolutely continuous.
3.3 Differential games and HamiltonJacobiIsaacs equations in metric spaces
This project is concerned with a gamebased interpretation of HamiltonJacobiIsaacs equations in metric spaces. We construct a twoperson continuoustime game in a geodesic space and show that the value function, defined by an explicit representation formula, is the unique solution of the HamiltonJacobi equation. Our result develops, in a general geometric setting, the classical connection between differential games and the viscosity solutions to possibly nonconvex HamiltonJacobi equations.
3.4 Asymptotically mean value harmonic functions in doubling metric measure spaces, subriemannian and RCD settings
We consider weakly and strongly asymptotically mean value harmonic (amvharmonic) functions on metric measure spaces which, in the classical setting, are known to coincide with harmonic functions. We show that the strongly amvharmonic functions are Hölder continuous for any exponent below one. More generally, we define the class of functions with finite amvnorm and show that functions in this class belong to a fractional Hajłasz–Sobolev space and their blowups satisfy the meanvalue property. Furthermore, in the weighted Euclidean setting we find an elliptic PDE satisfied by amvharmonic functions. We demonstrate that, in noncollapsed RCDspaces with vanishing metric measure boundary, Cheeger harmonic functions are weakly amvharmonic and that, in Carnot groups, weak amvharmonicity equivalently characterizes harmonicity in the sense of the subLaplacian. Moreover, in the homogeneous Carnot groups of step 2, we prove a Blaschke–Privaloff–Zaremba type theorem which yields the equivalence of both weak and strong amvharmonicity with harmonicity in the sense of the subLaplacian. Furthermore, similar results are discussed in the settings of Riemannian manifolds and for the Alexandrov surfaces.
4. Publications
4.1 Journals

Absolutely continuous mappings on doubling metric measure spaces, (P. Lahti and X. Zhou), submitted, arXiv

Quasiconformal and Sobolev mappings in nonAhlfors regular metric spaces when p>1, (P. Lahti and X. Zhou), submitted, arXiv

Quasiconformal and Sobolev mappings in nonAhlfors regular metric spaces, (P. Lahti and X. Zhou), submitted, arXiv

Differential games and HamiltonJacobiIsaacs equations in metric spaces, (Q. Liu), to appear in Minimax Theory and its Applications.
 T. Adamowicz, A. Kijowski, E. Soultanis, Asymptotically mean value harmonic functions in doubling metric measure spaces, submitted
6. T. Adamowicz, A. Kijowski, E. Soultanis, Asymptotically mean value harmonic functions in subriemannian and RCD settings, submitted
4.2 Books and other onetime publications
Nothing to report
4.3 Oral and Poster Presentations
*Talks of unit memebers at OIST are not included in this list.

PDE seminar, Hong Kong University of Science and Technology (zoom), X. Zhou, March 25, 2022

Himeji conference of Partial Differential Equation, Himeji (zoom), X. Zhou, March 24, 2022

Geometric and functional inequalities and applications seminar, zoom, X. Zhou, Feb 21, 2022

Riemann surfaces and Related topics conference, Osaka City University (zoom), X. Zhou, Feb 1315, 2022

Differential Geometry and Geometric Analysis Seminar, Princeton University (zoom), X. Zhou, November 10, 2021

Applied Analysis Seminar, The University of Tokyo (zoom), X. Zhou, October 28, 2021

The Mathematical Scociety of Japan Autumn Meeting (Geometry Section), X. Zhou, Sep 1417, 2021

Institute of Mathematics Polish Academy of Sciences Geometric function and mapping theory seminar (online), A. Kijowski, March 7, 2022
5. Intellectual Property Rights and Other Specific Achievements
Nothing to report
6. Meetings and Events
6.1 【Mini Course I】Curvature and Optimal transport
 Date: March 8 ~March 10, 2022
 Venue: OIST Campus Lab 4 and online
 Speaker: Professor Asuka Takatsu (Tokyo Metropolitan University)
6.2 【Mini Course II】 A brief introduction to branched optimal transport
 Date: March 10 ~March 14, 2022
 Venue: OIST Campus Lab 4 and online
 Speaker: Professor Jun Kitagawa (Michigan State University)
6.3 【Seminar】Natural \(p\)means for the \(p\)Laplacian in Euclidean space and the Heisenberg Group
 Date: May 14, 2021
 Venue: OIST Campus Lab 4 and online
 Speaker: Juan Manfredi (University of Pittsburgh)
6.4 【Seminar】Uniformization of weighted Gromov hyperbolic spaces and uniformly locally bounded geometry
 Date: May 28, 2021
 Venue: OIST Campus Lab 4 and online
 Speaker: Nageswari Shanmugalingam (University of Cincinnati)
6.5 【Seminar】Microlocal analysis of dplane transform on the Euclidean space
 Date: June 9, 2021
 Venue: OIST Campus Lab 4 and online
 Speaker: Hiroyuki Chihara (University of the Ryukyus)
6.6 【Seminar】Localization and isoperimetric inequalities
 Date: June 25, 2021
 Venue: OIST Campus Lab 4 and online
 Speaker: Shinichi Ohta (Osaka University and RIKEN)
6.7 【Seminar】System identification through Lipschitz regularized deep neural networks
 Date: July 16, 2021
 Venue: OIST Campus Lab 4 and online
 Speaker: Elisa Negrini (Worcester Polytechnic Institute)
6.8 【Seminar】Lipschitz mappings, metric differentiability, and factorization through metric trees
 Date: Auguest 11, 2021
 Venue: OIST Campus Lab 4 and online
 Speaker: Piotr Hajlasz (University of Pittsburgh)
6.9 【Seminar】Curve shrinking flow in Carnot groups
 Date: Auguest 27, 2021
 Venue: OIST Campus Lab 4 and online
 Speaker: Luca Capogna (Smith College)
6.10 【Seminar】On weak solutions to firstorder discount mean field games
 Date: October 8, 2021
 Venue: OIST Campus Lab 4 and online
 Speaker: Hiroyoshi Mitake (University of Tokyo)
6.11 【Seminar】Quasiconformal and Sobolev mappings in metric measure spaces
 Date: October 14, 2021
 Venue: OIST Campus Lab 4 and online
 Speaker: Panu Lahti (Chinese Academy of Sciences)
6.12 【Seminar】Fractals and the dynamics of Thurston maps
 Date: November 19, 2021
 Venue: OIST Campus Lab 4 and online
 Speaker: Mario Bonk (UCLA)
6.13 【Seminar】HelgasonFourier analysis techniques on hyperbolic spaces and sharp geometric and functional inequalities
 Date: December 10, 2021
 Venue: OIST Campus Lab 4 and online
 Speaker: Guozhen Lu (University of Connecticut)
6.14 【Seminar】Supercaloric functions for the parabolic pLaplace equation in the fast diffusion case
 Date: March 8, 2022
 Venue: online
 Speaker: Juha Kinnunen (Aalto University)
6.15 【CatchAll Colloquium】An invitation to NewtonOkounkov bodies ... and beyond.
 Date: October 21, 2021
 Venue: online
 Speaker: Megumi Harada（McMaster University）
6.16【CatchAll Colloquium】The GreenTao theorem for number fields
 Date: November 24, 2021
 Venue: online
 Speaker: Masato Mimura (Tohoku University)
6.17 【CatchAll Colloquium】Toward an Inclusive Research Environment
 Date: December 13, 2021
 Venue: online
 Speaker: Shun Yanashima (Tokyo Metropolitan University)
6.18 【CatchAll Colloquium】Goldbach’s Conjecture and the Riemann Hypothesis in Number Theory, and Their Relations to Zeta Functions
 Date: January 19, 2022
 Venue: online
 Speaker: Ade Irma Suriajaya (Kyushu University)
6.19 【CatchAll Colloquium】MongeAmpère equations related to optimal transport and geometric optics
 Date: February 22, 2022
 Venue: online
 Speaker: Jun Kitagawa (Michigan State University)
7. Other
Nothing to report.