Seminars-Liu
Organizers: 刘世平 (Shiping Liu, USTC)、孙俊 (Jun Sun WHU)、朱晓宝 (Xiaobao Zhu RUC)
Meeting 12: 11:00-12:00 December 23, 2025
Title: Global \(C^{1,\alpha}\) regularity for Monge-Ampère equations
Speaker: Jiakun Liu (刘佳堃) The University of Sydney
Zoom: 820-612-310-01 Code: 595143
Abstract: In this talk, we will discuss the global Hölder gradient estimate for solutions to the Dirichlet problem of the Monge-Ampère equation on strictly convex but not uniformly convex domains. This is a recent joint work with Qing Han and Yang Zhou.
Meeting 11: 15:00-16:30 December 09, 2025
Title: The decomposition of curvature measures via Hausdorff measures
Speaker: Baocheng Zhu (朱保成) 陕西师范大学
Tecent Meeting ID: 256 235 225 Code: 795559
Abstract: We will talk about the decomposition formulas for the curvature measures of convex bodies. The decomposition of a curvature measure of a convex body is with respect to Hausdorff measures of different dimensions restricted to the singular sets of the boundary of the convex body. The density functions and singular measures in the decomposition are explicitly given in terms of integrals of functions of the generalized curvatures of the convex body.
Meeting 10: 14:30-16:40 September 27, 2023
Title: Regularity for affine hyperbolic sphere and related problems
Speaker: Huaiyu Jian (简怀玉) 清华大学
Tecent Meeting ID: 512-5543-6492 Code: 202310
Abstract: We will talk on the recent results on the boundary regularity of affine hyperbolic sphere and related Monge-Ampere equations. We will give a sketch of the proof of the global analyticity of the affine hyperbolic sphere.
Title: Regularity of the chord log-Minkowski problem
Speaker: Jian Lu (鲁 建) 华南师范大学
Tecent Meeting ID: 512-5543-6492 Code: 202310
Abstract: The chord log-Minkowski problem arises from integral geometry, which was initially proposed by Lutwak-Xi-Yang-Zhang recently. In the smooth case, it is equivalent to solving a type of non-local Monge-Ampere equation on the unit hypersphere. Actually, it involves a Riesz potential defined on abounded domain. We will mainly talk about a new result on the regularity of solutions to the chord log-Minkowski problem, which is based on a joint work with Jinrong Hu and Yong Huang.
Meeting 9: 15:00-17:10 June 07, 2023
Title: The interior \(C^2\) estimate of Hessian type equations
Speaker: Chuanqiang Chen (陈传强) 宁波大学
Tecent Meeting ID: 470-635-247 Code: 202309
Abstract: In this talk, we introduce some results of interior estimate of Hessian type equations, including the k-Hessian equations and mixed Hessian type equations.
Title: k-Hessian方程的Green函数的正则性与估计
Speaker: Xinan Ma (麻希南) 中国科学技术大学
Tecent Meeting ID: 470-635-247 Code: 202309
Abstract: 我们引进两个辅助函数，得到一致梯度估计与二阶导数估计。利用这两个估计我们得到具有相应二阶估计的k-Hessian方程的Green函数存在性与渐进性。这是与高正焕和张德凯的合作工作。
Meeting 8: 14:00-16:10 May 29, 2023
Title: Rigidity of positively curved steady Ricci solitons on manifolds and orbifolds
Speaker: Yuxing Deng (邓宇星) 北京理工大学
Tecent Meeting ID: 381-502-719 Code: 202308
Abstract: Steady Ricci solitons are important examples of singularities models. In higher dimensions, singularity models can be steady Ricci solitons on orbifolds. In this talk, we will review some rigidity theorems on positively curved steady Ricci solitons on manifolds. We will also classify noncollapsed steady Ricci solitons on orbifolds with compact singularity of codimension \(2\), positive curvature operator and linear curvature decay.
Title: Equivariant KR flow and its application
Speaker: Xiaohua Zhu (朱小华) 北京大学
Tecent Meeting ID: 381-502-719 Code: 202308
Abstract: I will talk about a recent work jointly with F. Wang on equivariant KR flow via the Tian's partial \(C^0\)-estimate. As an application, we study the deformation of Kaehler metrics evolved in the KR flow on a G-spherical Fano manifold \(X\). We prove the limit of flow has a structure of \(G\)-spherical \(Q\)-Fano variety which can be realized as a \(C^*\)-degeneration of \(X\) induced by an element of Lie algebra of Cartan torus of \(G\).
Meeting 7: 15:00-17:10 May 15, 2023
Title: Capillary hypersurfaces
Speaker: Guofang Wang (王国芳) Albert-Ludwigs-Universitat Freiburg
Tecent Meeting ID: 402-679-295 Code: 202307
Abstract: Capillary hypersurfaces arise from many problems in mathematics and physics. By continuing a joint work with Xia Chao on capillary hypersurfaces supported on the unit sphere, we will focus in this talk on its counterpart on the half space. Various optimal geometric inequalities have been established by naturally generalizing previous methods. The talk bases on joint work with Xia Chao, Weng Liangjun and others.
Title: Pinching phenomena of Legendrian submanifolds in the unit sphere
Speaker: Yong Luo (罗勇) 重庆理工大学
Tecent Meeting ID: 402-679-295 Code: 202307
Abstract: In this report we will talk about pinching phenomena of Legendrian submanifolds in the unit sphere. In particular, utilizing the maximum principle to the Simons' type identity to minimal Legendrian submanifolds in the unit sphere, we obtain a new characterization of the minimal Calabi product Legendrian immersion of the totally geodesic Legendrian sphere and a point. This is based on joint works with Prof. Linlin Sun and Prof. Jiabin Yin.
Meeting 6: 15:00-17:10 May 08, 2023
Title: Some geometric and analytic results on Dirac operators
Speaker: Qun Chen (陈群) 武汉大学
Tecent Meeting ID: 329-976-434 Code: 202306
Abstract: Dirac operator was introduced by Paul Dirac as a "square root" of the Laplacian operator, it has deep relationship with the geometry and topology of manifolds. In this talk, we will introduce some geometric and analytic results on several topics related to Dirac operators, including Dirac equations, submanifold Dirac operators and Dirac-harmonic maps.
Title: Rigidity theorems of spacelike entire self-shrinking graphs in the pseudo-Euclidean space
Speaker: Linlin Sun (孙林林) 武汉大学
Tecent Meeting ID: 329-976-434 Code: 202306
Abstract: I will talk about some rigidity results of spacelike self-shrinkers. We firstly establish a new volume growth estimate for spacelike entire graphs in the pseudo-Euclidean space. Then by using this volume growth estimate and the co-area formula, we prove various rigidity results for spacelike entire self-shrinking graphs.
Meeting 5: 15:00-17:10 April 24, 2023
Title: Rigidity and non-rigidity of \(H^n/Z^{n-2}\) with scalar curvature bounded from below
Speaker: Yuguang Shi (史宇光) 北京大学
Tecent Meeting ID: 514-282-099 Code: 202305
Abstract: We present a counterexample to a generalization of Min-OO's hyperbolic rigidity theorem proposed by M. Gromov, and also prove a rigidity result of ALH manifolds with scalar curvature bounded from below. This talk is based on my recent joint work with my postdoc Y. H. Hu and my Ph. D. students P. Liu. T. Z. Hao.
Title: Rigidity of entire self-shrinking solutions to geometric flows
Speaker: Wenlong Wang (王文龙) 南开大学
Tecent Meeting ID: 514-282-099 Code: 202305
Abstract: In this talk, I will review some classical and introduce some new rigidity results of entire self-shrinking solutions to the Kahler-Ricci flow, the mean curvature flow, and the J-flow.
Meeting 4: 9:00-11:10 April 19, 2023
Title: Potential theory in conformal geometry
Speaker: Jie Qing (庆杰) University of California Santa Cruz
Tecent Meeting ID: 567-482-580 Code: 202304
Abstract: In this talk we would like to report my recent joint work with Shiguang Ma on the application of the potential theory in conformal geometry. We will mention many interesting equations in conformal geometry and the potential-theoretic approach to study them. In particular, we present the recent work on the extensions of Huber type theorem in higher dimensions under integral conditions of various curvature. We will demonstrate main ideas via the outlines of a proof of Huber Theorem.
Title: Conformal metrics with prescribed curvature functions
Speaker: Shiguang Ma (马世光) 南开大学
Tecent Meeting ID: 567-482-580 Code: 202304
Abstract: The Scalar curvature can be regarded as the trace of Schouten tensor, and is related to Yamabe equation. As a generalization, we consider the fully nonlinear equation of prescribed \(\sigma_k\) function of Schouten tensor, with Dirichlet boundary condition. I will talk about the history of this problem, the basic approach to proving the existence of the solutions and the recent progress we are making. This is a joint work with Jie Qing and Jinju Xu.
Meeting 3: 14:00-16:10 April 12, 2023
Title: Existence of mean curvature flow singularities
Speaker: Xiaoli Han (韩小利) 清华大学
Tecent Meeting ID: 888-117-250 Code: 202303
Abstract: Velazquez constructed a countable collection of mean curvature flow solutions in \(R^N, N\geq 8\). Each of these solutions becomes singular in finite time at which time the second fundamental form blows up. Guo and Sesum, Stolarski studied the behavior of the mean curvature of Velazquez's solution. While the construction provided by Velazquez's yields complete, non-compact mean curvature flow solutions. We will construct closed mean curvature flow solutions which become singular in finite time.
Title: The isoperimetric problem in the Riemannian manifold admitting a non-trivial conformal vector field
Speaker: Shujing Pan (潘淑婧) 中国科学技术大学
Tecent Meeting ID: 888-117-250 Code: 202303
Abstract: In this talk, we will discuss the isoperimetric problem by introducing a mean curvature type flow in the Riemannian manifold endowed with a non-trivial conformal vector field. This flow preserves the volume of the bounded domain enclosed by a star-shaped hyper-surface and decreases the area of hypersurface under certain conditions. We will prove the all-time existence and convergence of the flow. As a result, the isoperimetric inequality for such a domain is established. Especially, we solve the isoperimetric problem for the star-shaped hypersurface in the Riemannian manifold endowed with a closed, non-trivial conformal vector field, a wide class of warped product spaces is included. This is a joint work with Prof. Jiayu, Li.
Meeting 2: 14:00-16:10 March 2,7 2023
Title: 薛定谔流的诺依曼边值问题
Speaker: Youde Wang (王友德) 中国科学院数学与系统科学研究院、广州大学
Tecent Meeting ID: 549-825-116 Code: 202302
Abstract: 我们将回顾作为薛定谔流的物理背景的Landau-Lifshitz方程及其相关方程的历史，及其此方程与物理学(微电子学)、材料科学、流体力学的紧密联系。另一方面，也回顾此类方程与微分几何与拓扑学之间自然的联系。最后，介绍我们最近就薛定谔流(Landau-Lifshitz方程)的初始-诺依曼边值问题的强解及光滑解的存在性所取得的进展。
Title: On the fill-ins with scalar curvature bounded from below
Speaker: Guodong Wei (魏国栋) 中山大学
Tecent Meeting ID: 549-825-116 Code: 202302
Abstract: A triple of (generalized) Bartnik data \((Σ^{n-1},\gamma,H)\) consists of an orientable closed null-cobordant Riemannian manifold \((Σ^{n-1},\gamma)\) and a given smooth function \(H\) on \(Σ^{n-1}\). One basic problem in Riemannian geometry is to study: Under what kind of conditions does the Bartnik data \((Σ^{n-1},\gamma,H)\) admit a fill-in metric \(g\) with scalar curvature bounded below by a given constant? In this talk, we will discuss some estimates on the mean curvature of fill-ins with scalar curvature bounded from below, This is mainly based on joint work with Wenlong Wang at Nankai University.
Meeting 1: 14:00-16:10 March 01, 2023
Title: Compactness and singularity related to harmonic maps in higher dimension
Speaker: Jiayu Li (李嘉禹) 中国科学技术大学
Tecent Meeting ID: 660-218-866 Code: 202301
Abstract: In the talk we will review compactness results and regularity theorems of harmonic maps in higher dimension. We will review Evans and Bethuel partial regularity, Lin theorem, Riviere estimates, and our results in this field, and it's applications to triholomorphic maps between hyperkahler manifolds. At last, we will review harmonic maps related to black holes.
Title: Prescribing scalar curvatures: the negative case
Speaker: Chaona Zhu (朱超娜) Università degli Studi di ROMA “Tor Vergata”(罗马第二大学)
Tecent Meeting ID: 660-218-866 Code: 202301
Abstract: The problem of prescribing conformally the scalar curvature on a closed manifold of negative Yamabe invariant is always solvable, if the function to be prescribed is strictly negative, while sufficient and necessary conditions are known in the case that function is non-positive. Still in the case of a negative Yamabe invariant, Rauzy showed solvability, if the function to be prescribed is not too positive. In this talk we will review these results variationally, quantify the principal existence result of Rauzy and show under additional assumptions, that for a sign changing prescribed function solutions to the conformally prescribed scalar curvature problem, while existing, are not unique. In collaboration with Martin Mayer.
