Organizers

2022 Schedule

Date Speaker Title
Apr 12th, 2pm-3pm Chao Xia (Xiamen Univ.) CMC hypersurfaces in warped products: rigidity and quantitative stability
Apr 13th, 10am-11am Genggeng Huang (Fudan Univ.) Regularity of free boundary for the Monge-Ampere obstacle problem
Apr 20th, 10am-11am Xi-Nan Ma (USTC) Obata型积分恒等式的几个应用
Apr 27th, 10am-11am Wei Wei (Nanjing Univ.) A Liouville’s theorem for some Monge-Ampere type equations
May 11th, 10am-11am Yongsheng Zhang (Tongji Univ.) On the non-existence of solutions to the Dirichlet problem of minimal graphs
May 18th, 10am-11am Xiangwen Zhang (UCI) A geometric flow for Type IIA superstrings.
May 25th，9am-10am Hengyu Zhou (Chongqing Univ.) The Dirichlet problem of prescribed mean curvature equations via blow ups, I: monotone cases
June 1st, 9am-10am Connor Mooney (UCI) Some regularity questions for the special Lagrangian equation
June 8th, 3pm-4pm Jerome Wettstein (ETH) Half-Harmonic Gradient flow - Existence, Uniqueness and Regularity
June 15th, 10am-11am Youquan Zheng (Tianjin Univ.) Infinite time bubbling for the \(SU(2)\) Yang-Mills heat flow on \(\mathbb{R}^4\)
June 22nd, 10am-11am Shibing Chen (USTC) Singular set in the optimal transport problem
June 29th，9am-10am Siyuan Lu (McMaster) On the regularity of Lagrangian phase equation.
July 6th, 11am-12am Dekai Zhang (Shanghai Univ.) The exterior Dirichlet problem of homogeneous k-Hessian equations
July 13th, 9am-10am
July 13th, 10am-11am		 Yi Wang (JHU)
Pak Tung Ho (Sogang Univ.)		 Rigidity of local minimizers of the \(\sigma_k\) functional.
The weighted Yamabe problem

Abstracts

Pak Tung Ho

In this talk, I will talk about what the weighted Yamabe problem is and mentioned some related results that Jinwoo Shin (KIAS) and I obtained.

Yi Wang

In this talk, I will present a result on the rigidity of local minimizers of the functional \(\int \sigma_2+ \oint H_2\) among all conformally flat metrics in the Euclidean (n + 1)-ball. We prove the metric is flat up to a conformal transformation in some (noncritical) dimensions. We also prove the analogous result in the critical dimension n + 1 = 4. The main method is Frank-Lieb’s rearrangement-free argument. If minimizers exist, this implies a fully nonlinear sharp Sobolev trace inequality. I will also discuss a nonsharp Sobolev trace inequality. This is joint work with Jeffrey Case.

Dekai Zhang

We consider the regularity problem of solutions of homogeneous k-Hessian equations in an exterior domain in R^n. We show the \(C^{1,1}\) regularity of the solutions. The key ingredient is to prove \(C^2\) estimates for the approximating solutions, which are independent of the approximation. As an application, we prove an almost monotonicity formula along the level set of the the approximating solution. We also consider the corresponding regularity problem for the complex k-Hessian equation in \(C^n\). This is a joint work with Dr. Zhenghuan Gao and Prof. Xinan Ma.

Siyuan Lu

In this talk, I will first introduce the background and motivation for the study of Lagrangian phase equation. I will then discuss my recent work on the regularity of Lagrangian phase equation. In the second part, I will discuss some open problems relating to Lagrangian phase equation.

Shibing Chen

In the optimal transport problem, it is well known that the optimal map can be discontinuous when the target domain is non-convex. In this talk we will discuss some recent results and open problems about the structure of the singular set in the optimal transport problem with non-convex target domains.

Youquan Zheng

We investigate the long time behaviour of the Yang-Mills heat flow on the bundle \(\mathbb{R}^4\times SU(2)\). Waldron proved global existence and smoothness of the flow on a compact connected \(4\)-manifold, leaving open the issue of the behaviour at infinite time. We exhibit two types of long-time bubbling: first we construct an initial data and a globally defined solution which blows-up in infinite time at a given point in \(\mathbb R^4\). Second, we prove the existence of bubble-tower also in infinite time. This answers in definitive manner the basic properties of the heat flow of Yang-Mills connection in the critical dimension \(4\) and shows in particular that in general one cannot expect that this gradient flow converges to a Yang-Mills connection on \(4-\)manifolds. We emphasize that we do not assume for the first result any symmetry assumption; whereas the second result on the existence of the bubble-tower is in the \(SO(4)\)-equivariant class, but nevertheless new. This is a joint work with Yannick Sire and Juncheng Wei.

Jerome Wettstein

In this talk, we will discuss some of the results obtained in my PhD work pertaining to the half-harmonic gradient flow which is governed by the non-local PDE:\(\partial_t u + (-\Delta)^{1/2} u \perp T_u N,\) for functions \(u: [0,+\infty[ \times S^1 \to N \subset \mathbb{R}^{K}\). In particular, we shall address questions pertaining to existence of solutions, uniqueness in different senses and regularity as well as a preliminary investigation of bubbling while drawing connections with the nowadays quite classical harmonic gradient flow and the corresponding results proven by Struwe in the 1980’s.

Connor Mooney

The special Lagrangian equation is a fully nonlinear elliptic PDE whose solutions are potentials for volume-minimizing Lagrangian graphs. There exist continuous viscosity solutions to the Dirichlet problem for this equation, but many basic regularity questions (such as whether these solutions have bounded gradient) remain open. In this talk we will discuss some of these questions. We will use them to motivate the more general question of whether homogeneous functions with nowhere vanishing Hessian determinant can change sign when the degree of homogeneity is between zero and one, and we will answer this question in the negative.

Hengyu Zhou

This is the first one of our series of papers to study the Dirichlet problem of prescribed mean curvature equations via a blowup technique inspired from Shoen-Yau’s proof on the positive mass theorem. These equations includes Jang equations and minimal surface equations in warped product manifolds. We relate the solvability of these Dirichlet problems with a toplogical condition (NCf-condition) from the prescribed mean curvature functions. A key ingredient of our proof are an extension of curvature estimates of C^2 almost minimal boundaries from Simon’s idea.

Xiangwen Zhang

The equations of the flux compactifications of Type IIA superstrings were written down by Tomasiello and Tseng-Yau. To study those equations, we introduce a natural geometric flow on symplectic Calabi-Yau 6-manifolds. In this talk, we will discuss the recent progress on the study of the Type IIA flow. This is based on a joint project with Fei, Phong and Picard.

Yongsheng Zhang

We will introduce some developments about the Dirichlet problem of minimal graphs in Euclidean spaces, establish an Allard boundary regularity in this setting, and apply that to obtain several new results on the non-existence of solutions.

Wei Wei

In this paper we study a Monge-Ampere type equation that interpolate the classical 2-Yamabe problem in conformal geometry and the 2-Hessian equation in dimension 4. This is a joint work with Hao Fang and Biao Ma.

Xi-Nan Ma

我们回忆Obata（1971JDG）型积分恒等式，它已经有50多年的历史。在半线性椭圆方程上Gidas-Spruck于1981年开始应用。在P-Laplace方程的研究中Serrin-Zou于2002年开始发展其相关技巧，在CR几何与Heisenberg群上由Jerison-Lee于1988年开始应用。我们将报告近来在后两方面的工作。

Genggeng Huang

In this talk, we talk about the regularity of the free boundary in the Monge-Ampere obstacle problem \[\begin{equation} \begin{split} \det D^2 v= f(y)\chi_{\{v>0\}},\quad \text{in}\quad \Omega\\v=v_0,\quad \text{on}\quad \partial\Omega. \end{split} \end{equation}\] Assume that \(\Omega\) is a bounded convex domain in \(\Bbb R^n\), and \(f, v_0>0\).Then \(\Gamma=\partial \{v=0\}\) is smooth if \(f\) is smooth; and \(\Gamma\) is analytic if \(f\) is analytic. This is a joint work with Prof. Tang Lan and Prof. Wang Xu-Jia.

Chao Xia

Brendle proved Alexandrov’s theorem that classified closed embedded constant mean curvature (CMC) hypersurfaces in certain warped products. In joint works with Guohuan Qiu and Junfang Li, among others, we established Reilly type integral formula to reprove Brendle’s result. In this talk, we introduce a recent joint work with Julian Scheuer, to establish quantitative stability for closed embedded almost CMC hypersurfaces in warped products, which is based on Li-Xia’s new proof of Brendle’s result and Scheuer’s rigidity-to-stability criteria.


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