Workshop on Geometric Analysis and PDEs

Time：June 6-8, 2025

Venue：North China Hotel

Agenda：

Abstract：

Speaker: Ben Andrews, The Australian National University

Host: Weimin Sheng, Zhejiang University

Title: Self-similar solutions of Gauss curvature flow

Abstract: I will discuss recent work with Devesh Rajpal on hypersurface flows by a power $\alpha<\frac{1}{n+2}$ of Gauss curvature in $\RR^{n+1}$.  We prove a convexity property of the Gauss curvature flow entropy along the action of the special affine group, and use this to derive an a priori isoperimetric bound for self-similar solutions and prove existence of new nontrivial self-similar solutions using a modified Gauss curvature flow.

Speaker: Jiyuan Han, Westlake University

Host: Shibing Chen, University of Science and Technology of China

Title: On the existence of weighted-cscK metrics

Abstract: Weight-cscK metrics provide a universal framework for the study of canonical metrics, e.g, extremal metrics, Kahler Ricci soliton metrics, \mu-cscK metrics. In joint works with Yaxiong Liu, we prove that on a Kahler manifold X, the G-coercivity of weighted Mabuchi functional implies the existence of weighted-cscK metrics. In particular, there exists a weighted-cscK metric if X is a projective manifold that is weighted K-stable for models. At the end of the talk, I will report some progress on singular varieties.

Speaker: Ge Xiong, TongJi University

Host: Yong Huang, Hunan University

Title: A matroid polytope approach to sharp affine isoperimetric inequalities

Abstract: New sharp affine isoperimetric inequalities for volume decomposi-tion functionalsare established. To attack these extremal problems, we find the recursion formulas of volume decomposition functionals and prove their domain is the relative interior of a matroid polytope. Applications of matroid polytopes in convex geometry are provided. This talk is based on the joint work with Liu Yu-de and Sun Qiang.

Speaker: Robert McCann, University of Toronto

Host: Thierry De Pauw, Westlake University

Title: The monopolist's free boundary problem in the plane: an excursion into the economic value of private information

Abstract: The principal-agent problem is an important paradigm in economic theory for studying the value of private information: the nonlinear pricing problem faced by a monopolist is one example; others include optimal taxation and auction design. For multidimensional spaces of consumers (i.e. agents) and products, Rochet and Chone (1998) reformulated this problem as a concave maximization over the set of convex functions, by assuming agent preferences are bilinear in the product and agent parameters. This optimization corresponds mathematically to a convexity-constrained obstacle problem. The solution is divided into multiple regions, according to the rank of the Hessian of the optimizer.

Apart from four possible pathologies, if the monopolists costs grow quadratically with the product type we show that a smooth free boundary delineates the region where it becomes efficient to customize products for individual buyers. We give the first complete solution of the problem on square domains, and discover new transitions from unbunched to targeted and from targeted to blunt bunching as market conditions become more and more favorable to the seller.

Based on work with Cale Rankin (Monash University) and Kelvin Shuangjian Zhang (Fudan University) https://arxiv.org/abs/2412.15505

Speaker: Lizhen Ji

Host: Min-Chun Hong, The University of Queensland

Title: The Uniformization Theorem: History, Proofs, and Lasting Impacts

Abstract: The uniformization theorem is one of the most significant results in mathematics, with a deceptively simple statement familiar to many: every simply connected Riemann surface is biholomorphic to one of three standard surfaces—the Riemann sphere, the complex plane, or the open unit disk. This theorem has profound extensions to higher dimensions, including Thurston’s geometrization program and, notably, the resolution of the Poincaré conjecture. It is no wonder that this theorem is one of the most important theorems, if not the most important theorem, in the last 150 years.

However, its rich history and far-reaching impacts are often underappreciated. For example, the initial formulations and attempted proofs by Klein and Poincaré using the method of continuity introduced many original ideas that have not been fully explored. Later, Teichmüller's innovative use of the method of continuity played a crucial role in his revolutionary work on Teichmüller space, profoundly influencing the study of moduli spaces of Riemann surfaces. Yet, this deep historical and conceptual connection is frequently overlooked.

In this talk, I will trace the historical development of the uniformization theorem, from its early formulations to its lasting influence across mathematics. I will explore its connections to seemingly unrelated results and modern mathematical areas, particularly in Teichmüller theory and moduli spaces. Through this exploration, I aim to shed light on the theorem's profound influence, inspire new perspectives on its unifying role in contemporary mathematics.

Speaker: James McCoy, Newcastle University

Host: Jiakun Liu, The University of Sydney

Title: Curve diffusion flows in cones

Abstract: We study families of smooth, embedded, regular planar curves with generalised Neumann boundary conditions inside cones, satisfying three variants of the fourth-order nonlinear curve diffusion flow: curve diffusion flow with length penalisation and two forms of constrained curve diffusion flow with fixed length. Assuming neither end of the evolving curve reaches the cone tip, existence of smooth solutions for all time given quite general initial data is well known for the constrained flows is well known, but classification of limiting shapes is generally not known. We prove for the constrained flows smooth exponential convergence of solutions in the $C^\infty$-topology to a circular arc centred at the cone tip with the same length as the initial curve. In the length penalised case, we show smooth exponential convergence under suitable rescaling to a circular arc centred at the cone tip. This is joint work with Mashniah Gazwani.

Speaker: Changwei Xiong, Sichuan University

Host: Yong Wei, University of Science and Technology of China

Title: On Escobar-type results for Steklov eigenvalue problems

Abstract: In 1999, J. Escobar proposed the conjecture: for an n-dimensional (n>=3) compact Riemannian manifold with boundary with nonnegative Ricci curvature and boundary principal curvatures bigger than or equal to c>0, the first nonzero Steklov eigenvalue of the manifold is no less than c, with the equality only for a Euclidean ball of radius 1/c. In the talk we will discuss results related to this conjecture, both for Steklov problems on functions, and for Steklov problems on differential forms. Part of recent results I shall present are joint with Qinyong Liu.

Speaker: Florica Cirstea, The University of Sydney

Host: Maolin Zhou, Nankai University

Title: Existence and classification of solutions for nonlinear elliptic equations with singular potentials

Abstract: In this talk, we present new results on the existence of and classification of the behaviour near zero of the positive $C^2(B_R(0)\setminus \{0\})$ solutions of $$ -\Delta u = \frac{u^{2^\star(s)-1}}{|x|^s} - \mu \frac{u^q}{|x|^\tau} \quad \text{in } B_R(0)\setminus\{0\}, $$ where $B_R(0)$ is the open ball in $\mathbb R^n$ centered at $0$ and of radius $R>0$.

Here, $n\geq 3$, $q>1$, $\mu>0$, $\tau\in \mathbb R$, $s\in (0,2)$ and $2^*(s)=2(n-s)/(n-2)$. The classification of singularities is intimately connected with the position of $\tau$ with respect to $2$, as well as $s$.

We will focus on the case of removable singularities to emphasize the changes between $\tau<2$ and $\tau\geq 2$. We will also examine certain singular asymptotic profiles using Pohozaev type arguments.

We use a dynamical systems approach to obtain the existence of solutions with the desired profile.