主辦單位：中國科學院數學與系統科學研究院

協辦單位：中國科學技術大學幾何與物理中心

時間：2021年12月4-5日

騰訊會議：942 663 0176 會議密碼：2021

會議日程

題目: The J-equation and deformed Hermitian-Yang-Mills equation

報告人: 陳杲 (中國科學技術大學)

時間: 2021.12.04 8:30am-9:20am

摘要: The deformed Hermitian-Yang-Mills (dHYM) equation is the mirror equation for the special Lagrangian equation. The “small radius limit” of the dHYM equation is the J-equation, which is closely related to the constant scalar curvature K？hler (cscK) metrics. In this talk, I will explain my recent result that the solvability of the J-equation is equivalent to a notion of stability. I will also explain my similar result on the supercritical dHYM equation as well as the application of my results to the cscK problem.

題目: The Bach tensor, Bach flows, and special metrics on high symmetry 4-manifolds

報告人: Brian Weber (上海科技大學)

時間: 2021.12.04 9:40am-10:30am

摘要: Using an efficient computational framework, we find simple expressions for metric tensors on certain cohomogeneity-1 manifolds, up through the Bach tensor. This allows complete characterization of the special metrics: except for some B^t-flat metrics, all special metrics are conformal to the extremal K？hler metrics, which has a 3-dimensional moduli space. We give the Bach tensor a novel dynamical interpretation in terms of the “Bach scalar”. We find long-time solutions to the 4th order dissipative flow gt = 4Bach. After a double DeTurck trick—one for diffeomorphism gauge fixing and the other for conformal gauge fixing—we find the Bach flow transforms into another gradient flow: the gradient flow of the Bach scalar, which, remarkably, is a linear flow. This flow is analogous to the 4th order Calabi flow, but easier to solve.

題目: The dissolving limit and large volume limit of Einstein-Bogomol’nyi metrics

報告人: 姚成建 (上海科技大學)

時間: 2021.12.04 10:50am-11:30am

摘要: Einstein-Bogomol’nyi metrics, which physically models the Cosmic Strings, solves the Einstein’s Fields Equation coupled with an Abelian gauge field and a Higgs field. In this talk,

I will present one existence theorem for Einstein-Bogomol’nyi metrics. I will also discuss the behaviors of the metrics as the volume approaches the lower bound and infinity respectively. Part of this talk is based on the joint work with Garcia-Fernandez and Pingali.

題目: Asymptotic behaviors of nonlinear dispersive equations

報告人: 趙立豐 (中國科學技術大學)

時間: 2021.12.04 2:00pm-2:50pm

摘要: Nonlinear dispersive equations arise in many physical background such as quantum mechanics, fluids and optics. In this talk, we will focus on the works on asymptotic behaviors including soliton resolution and blowup phenomena. Some of our recent results will be mentioned.

題目: On geodesic orbit spaces with intermediate subgroups

報告人: 陳慧斌（南京师范大學）

時間: 2021.12.04 3:10pm-4:00pm

摘要: Assume H, K and G are compact connected Lie groups satisfying H ？ K ？ G. We first show that if the total space G/H is G-geodesic orbit, then the base space G/K and the fiber K/H with restricted metrics are all geodesic orbit. Furthermore, we will show how the representations of H and K affect the structures of g.o. metrics on G/H. In particular, we study the special case when K is a direct product of two Lie groups H and L. As an application, we determine all G-g.o. metrics on G/H arising from strongly isotropy irreducible spaces G/(H ×L).

題目: HCMU metrics and isometric immersion problems

報告人: 吴英毅（中国科學院大學）

時間: 2021.12.04 4:20pm-5:10pm

摘要: HCMU metric is 1-dimensional non-CSC singular extremal K？hler metric. In this talk, we will first review the definition of HCMU metric and some important properties. Then we will focus on some isometric immersion problems related to HCMU metrics.

題目: Huber’s theorem for conformally compact manifolds

報告人: 李宇翔（清华大學）

時間: 2021.12.05 8:30am-9:20am

摘要: Huber’s Theorem states that if (Σ, g) is a complete surface with K ？ 1 < +∞, then (Σ, g) is conformally equivalent to a closed surface with finitely many points removed. Such a result is not true for a higher dimensional manifold. In this talk, we will show that if (Mn , g) is conformally compact (i.e. (M, g) is conformally equivalent to a domain of a closed manifold), and if Ric is in L , then Huber’s theorem holds on M .

題目: Construction of concentrated solutions for some nonlinear equations

報告人: 楊軍 (广州大學)

時間: 2021.12.05 9:40am-10:30am

摘要: We will review some results on the construction of concentrated solutions for some

nonlinear equations. Besides some delicate techniches in analysis, the construction relies on the variational properties of limit set.

報告人: 羅勇 (重庆理工大學)

時間: 2021.12.05 10:50am-11:30am

題目: On energy gap phenomena of the Whitney spheres in a complex space form

摘要: In the theory of Lagrangian submanifolds of a complex space form, the Whitney spheres play a similar role as that of totally umbilical hypersurfaces in a real space form. In this talk

we introduce new higher order partial differential equations satisfied by the Whitney spheres and give new characterization of them as solutions to these equations which satisfy certain pointwise

or integral extrinsic curvature assumptions. This talk is based on joint works with Dr. Jiabin Yin and Dr. Liuyang Zhang.

報告人: 陳立（湖北大學）

時間: 2021.12.05 2:00pm-2:50pm

題目: Mixed Hessian equations on closed K？hler manifolds

摘要: In this talk, we consider a Hessian equation with its structure as a combination of elementary symmetric functions on closed K？hler manifolds. We provide a sufficient and necessary condition for the solvability of this equation, which generalizes the results of Hessian equation and Hessian quotient equation. The key to our argument is a clever use of the special properties

報告人: 馬元慶 (中國科學技術大學)

時間: 2021.12.05 3:10pm-4:00pm

題目: Ricci curvature integrals, local functionals, and the Ricci flow

摘要: Consider a closed Riemannian manifold (M, g) of dimension m ≥ 3 and the volume of M is the same as the standard sphere. If p > m/2 and the integral of the negative part of (Rc ？ (m ？ 1)g)p on manifold M is su？iciently small, we show that the normalized Ricci flow initiated from (M, g) will exist immortally and converge to the standard sphere, via estimating the localized Perelman functionals and the distance distortion along the Ricci flow.

報告人: 于江濤 (中國科學技術大學)

時間: 2021.12.05 4:20pm-5:10pm

題目: Complete 2-convex translating solitons to the mean curvature flow in Rn+1

摘要: Using a result of Derdzinski on Codazzi tensors, we extend the work of Spruck and Xiao in complete translating solitons to the mean curvature flow in R3 with nonnegative mean curvature to higher dimensions. More precisely, for n ≥ 3, we show that n-dimensional complete 2-convex translating solitons for the mean curvature flow in Rn+1 are convex.