# 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 sub-Riemannian 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 game-based interpretation of Hamilton-Jacobi-Isaacs 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 mini-courses 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 Grant-in-Aid for Early-Career Scientists (April 2022-March 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     January-April, 2022
• Rotation Graduate student, Jonas Schneider     January-April, 2022

## 2. Collaborations

### 2.1 Collaborators

• Panu Lahti, Chinese Academy of Sciences
• Projects:
1. Quasiconformal and Sobolev mappings in non-Ahlfors regular metric spaces
2. Absolutely continuous mappings on doubling metric measure spaces
• Qing Liu, Fukuoka University
• Projects:
1. Differential games and Hamilton-Jacobi-Isaacs equations in metric spaces
2. Hoorizontally quasiconvex envelope in the Heisenberg group
• Tomasz Adamowicz, Institute of Mathematics of the Polish Academy of Sciences and Elefterios Soultanis, Radboud University
• Projects:
1. 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 non-Ahlfors regular metric spaces

We show that a mapping f : X → Y satisfying the metric condition of quasiconformality outside suitable exceptional sets is in the Newton-Sobolev class $$f\in N^{1,1}_{loc}(X;Y)$$. Contrary to previous works, we only assume an asymptotic version of Ahlfors-regularity on X, Y . This allows many non-Ahlfors 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 Q-dimensional 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 Q-regularity, 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 Q-absolutely 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 Q-absolutely continuous mapping on an Ahlfors Q-regular 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 Radon-Nikodym property. Conversely, though a continuous Sobolev mapping$$N^{1,p}_{loc}(X;V)$$) is generally not locally Q-absolutely continuous, this implication holds if f is further assumed to be pseudomonotone. It follows that pseudomonotone mappings satisfying a relaxed quasiconformality condition are also Q-absolutely continuous.

### 3.3 Differential games and Hamilton-Jacobi-Isaacs equations in metric spaces

This project is concerned with a game-based interpretation of Hamilton-Jacobi-Isaacs equations in metric spaces. We construct a two-person continuous-time game in a geodesic space and show that the value function, defined by an explicit representation formula, is the unique solution of the Hamilton-Jacobi equation. Our result develops, in a general geometric setting, the classical connection between differential games and the viscosity solutions to possibly nonconvex Hamilton-Jacobi 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 (amv-harmonic) functions on metric measure spaces which, in the classical setting, are known to coincide with harmonic functions. We show that the strongly amv-harmonic functions are Hölder continuous for any exponent below one. More generally, we define the class of functions with finite amv-norm and show that functions in this class belong to a fractional Hajłasz–Sobolev space and their blow-ups satisfy the mean-value property. Furthermore, in the weighted Euclidean setting we find an elliptic PDE satisfied by amv-harmonic functions. We demonstrate that, in non-collapsed RCD-spaces with vanishing metric measure boundary, Cheeger harmonic functions are weakly amv-harmonic and that, in Carnot groups, weak amv-harmonicity equivalently characterizes harmonicity in the sense of the sub-Laplacian. 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 amv-harmonicity with harmonicity in the sense of the sub-Laplacian. Furthermore, similar results are discussed in the settings of Riemannian manifolds and for the Alexandrov surfaces.

## 4. Publications

### 4.1 Journals

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

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

3. Quasiconformal and Sobolev mappings in non-Ahlfors regular metric spaces, (P. Lahti and X. Zhou), submitted, arXiv

4. Differential games and Hamilton-Jacobi-Isaacs equations in metric spaces, (Q. Liu), to appear in Minimax Theory and its Applications.

5. 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 one-time publications

Nothing to report

### 4.3 Oral and Poster Presentations

*Talks of unit memebers at OIST are not included in this list.

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

2. Himeji conference of Partial Differential Equation, Himeji (zoom), X. Zhou, March 2-4, 2022

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

4. Riemann surfaces and Related topics conference, Osaka City University (zoom), X. Zhou, Feb 13-15, 2022

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

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

7. The Mathematical Scociety of Japan Autumn Meeting (Geometry Section), X. Zhou, Sep 14-17, 2021

8. 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 d-plane 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: Shin-ichi 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 first-order 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】Helgason-Fourier 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 p-Laplace equation in the fast diffusion case

• Date: March 8, 2022
• Venue:  online
• Speaker: Juha Kinnunen (Aalto University)

### 6.15 【Catch-All Colloquium】An invitation to Newton-Okounkov bodies ... and beyond.

• Date: October 21, 2021
• Venue:  online
• Speaker: Megumi Harada（McMaster University）

### 6.16【Catch-All Colloquium】The Green--Tao theorem for number fields

• Date: November 24, 2021
• Venue:  online
• Speaker: Masato Mimura (Tohoku University)

### 6.17 【Catch-All Colloquium】Toward an Inclusive Research Environment

• Date: December 13, 2021
• Venue:  online
• Speaker: Shun Yanashima (Tokyo Metropolitan University)

### 6.18 【Catch-All 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 【Catch-All Colloquium】Monge-Ampère 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.