**Guohuan Qiu**(Academy of Mathematics and Systems Science) qiugh[AT]amss.ac.cn**Liming Sun**(Academy of Mathematics and Systems Science) lmsun[AT]amss.ac.cn

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 |

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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。

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.

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.