2016年几何和几何分析国际会议会议报告题目及摘要简

“2016年几何和几何分析国际会议”会议报告题目及摘要简表

1. Rong Xiaochun

题目：Quantitative volume space form rigidity under lower Ricci curvature bound

摘要：Consider a compact $n$-manifold $M$ of Ricci curvature bounded below, normalized by $(n-1)H$, where $H=\pm 1$ or $0$. Then $M$ is isometric to a $H$-space form under either of the maximal volume conditions: (i) (Bishop) there is $\rho>0$ such that every $\rho$-ball on $M$ has the maximal volume i.e.,the volume of a $\rho$-ball in a simply connected $H$-space form; (ii) (Ledrappier-Wang)For $H=-1$, the volume entropy of $M$ is maximal i.e., the volume entropy of any hyperbolic$n$-manifold. We will discuss quantitative version of the two `volume' rigidityfor space form. This work is joint with Lina Chen and Shisheng Xu of Capital Normal University.

2. Xu Xingwang

题目：Negative gradient flow and scalar curvature functions

摘要：In this talk, I would like to describe an alternative approach to some quantitative results of Escobar and Schoen for metrics of prescribed scalar curvature on closed Riemannian manifolds with vanishing Yamabe invariant via a negative gradient curvature flow. The main result is to remove the so-called (n-3) flatness condition and conformal flatness condition proposed by Escobar and Schoen; hence answering a conjecture by Kadzan and Warner affirmatively. This talk is based on my joint work with Dr. Ngo Quoc Anh.

3. Ma Li

题目：Recent progress about ricci solitons

摘要：In this talk, we discuss the recent progress about gradient ricci solitons. We start from basic examples of ricci solitons and basic relations about potential functions and local volume controls. Then we review some recent results. We give partial proofs. We also mention related results about yamabe solitons and mean curvature solitons.

4. Bao Jiguang

题目：On Exterior Problems of Elliptic and Parabolic Monge-Ampere Equations

摘要：We shall survey the results on Jorgens-Calabi-Pogorelov theorem and the asymptotic behavior of elliptic Monge-Ampereequations. As the main content of this talk, we apply level set and nonlinear perturbation method to obtain the asymptotic behavior of the solution to a kind of parabolic Monge-Ampere equation at infinity. Jorgens-Calabi-Pogorelov theorem for parabolic Monge-Ampere equation can be regarded as a special case of our result.

5.Wu Jianhua

题目：Positive Solutions of the Competition Model in Unstirred Chemostat

摘要：In this talk, positive solutions of the unstirred chemostat model are given in the case of one and two resources, the structure of positive solutions are also studied with respect to the unstirred chemostat with inhibitors, and the related results are also surveyed if possible. The main ingredients include the global bifurcation theory, the fixed point index theory, perturbation method and simulations.

6.Zhou Chunqin

题目：Vanishing Pohozaev constant and removability of singularities 摘要：Conformal invariance of two-dimensional variational problems is a condition known to enable a blow-up analysis of solutions and to deduce the removability of singularities. In this paper, we identify another condition that is not only sufficient, but also necessary for such a removability of singularities. This is the validity of the Pohozaev identity.In situations where such an identity fails to hold, we introduce a new quantity, called the {\it Pohozaev constant}, which on one hand measures the extent to which the Pohozaev identity fails and on the other hand provides a characterization of the singular behavior of a solution at an isolated singularity. We apply this to the blow-up analysis for super-Liouville type equations on Riemann surfaces with conical singularities, because in the presence of such singularities,conformal invariance no longer holds and a local singularity is in general non-removable unless the Pohozaev constant is vanishing. This is a joint work with Juergen Jost and Zhu Miaomiao.

7. Qing Jie

题目：On non negatively curved hypersurfaces in hyperbolic space

摘要：We will report a proof of the conjecture of Alexander and Currier on the embeddedness of a non negatively curved hypersurfaces in hyperbolic spaces.

8. Jian Huaiyu

题目：Regularity Analysis for A Class of Singular Elliptic Equations 摘要：This talk is based on the recent joint works with Xu-Jia Wang and Jian Lu. We will study the optimal regularity for the Dirichlet problem to a Class of fully nonlinear Elliptic Equations with singular terms on the boundary. Moreover, we establish the asymptotic expansions near the boundary of solutions to the problem.

9. Xu Deliang

题目：Continuity of almost harmonic map with approximation in a critical Zygmmund class 摘要：We study the continuity of almost harmonic map (or approximation of harmonic maps) (),u A u u f ?=??+

()1,u H N ∈Ω, ,2n R n Ω?> a bounded domain and

(),N h compact Riemannian manifold with 2log n q

f L L ∈. The exponent 2n

is critical in the sense that even for linear equation ( the case (),0A u u ??=), we cannot expect the continuity of the solution. We derive a sharp condition to guarantee the continuity of solution u . In particular, we show that under some suitable condition (roughly speaking, BMO norm of u is sufficient small), the condition 12n q -> is sufficient for the continuity of map u , but not for 12n q -=.

Indeed, We consider continuity of solutions of the following more general elliptic system introduced by Rivi`ere

u u f ?=Ω??+

where is an anti-symmetric 1-form valued matrix.

10. Liu Huili

题目：三维仿射与半-欧氏空间中的曲线

摘要：研究二维、三维空间中的参数曲线的中心仿射微分几何。参数曲线的仿射性质、度量性质及其关系。给出一些特殊曲线。

11. Liu Baiyu

题目：Fundamental gap of the drifting Laplacian operator

摘要：In this talk, we survey some recent results on the estimate for the eigenvalue of Laplace operator. Then we consider the drifting Laplacian operator with zero Dirichlet boundary value. By generalizing some results of N. Korevaar and S.-T. Yau to gain a Hessian estimate of the first eigenfunction, we get a lower bound of the difference of the firast and second eigenvalues of the drifting Laplacian.

12. Wang Xiaoliu

题目：The nonlocal curvature flows

摘要：In this talk, some nonlocal planar cuvature flows will be introduced, including the area-preserving flow and the length-preserving one, etc. The locally convex initial curve could be embedded or immersed. The velocities of the flows could have various forms. Some sharp conditions on immersed initial curves are established to yield the singularity of the flow at a finte time. In addition, the symmetry of curves plays an important role in their evolution.

13. Cheng Liang

题目：On the Type IIb and Type III mean curvature fllow

摘要：In this talk, we first recall the definition of Type IIb and Type III mean curvature flows introduced by Hamiton. Then we show that the entire graphs satisfying certain conditions could be Type IIb. Our results lead us to getting the nontrivial examples for the Type IIb mean curvature flow. We also introduce the monotonicity formulas for themean curvature flow which are related to self-expanders. Then we use the monotonicity to study the asymptotic behavior of Type III mean curvature flow on noncompact hypersurfaces. Partial of this talk is the joint work with Natasa Sesum.

14. Cheng Xinyue

题目：The Existence of Einstein Finsler Metrics

摘要：In this talk, we study a class of Finsler metrics in the form F =__(_=_) de_ned by a Riemannian metric _ and a 1-form _. We _ndequations on _; _ and _ that characterize Einstein metrics and construct a family of Einstein metrics on S3 with Ric = 2F2, Ric = 0 and Ric = 2F2, respectively. This family of metrics provide an important class of Finsler metrics in dimension three, whose Ricci curvature is a constant,but the ag curvature is not.

15. Chen Fangwei

题目：V olume inequalities for asymmetric orlicz zonotopes

摘要：In this talk, we introduce the asymmetric Orlicz zonotopes by us-ing the method of shadow system. We will show the volume product inequality and volume ratio inequality for asymmetric Orlicz zonotopes, along with their equality cases.

16.Li Fengquan

题目：Steiner symmetric method of paratial differential equations

摘要：In this talk, a survey on the Steiner symmetric method of paratial differential equations will be given. First, we will recall some relusts about linear elliptic and parabolic problems via the

Steiner symmetric method. Then, some recent comparison results for solutions of a class of sublinear elliptic and parabolic equations by Steiner symmetrization will be shown.

17.Wang Chunpeng

题目：Smooth Transonic Flows of Meyer Type in De Laval Nozzles

摘要：This talk concerns smooth transonic steady potential flows of Meyer type in two-dimensional de Laval nozzles, which are governed by an equation of mixed type with degeneracy at the sonic state. For a $C^2$ transonic flow of Meyer type, it is shown that the set of exceptional points is a closed line segment (may be empty or only one point). Furthermore, a smooth transonic flow of Meyer type with nonexceptional points is unstable with respect to $C^0$-norm of the velocity for a $C^1$ small perturbation in the shape of the wall (even if the wall is still smooth). Then we seek a smooth transonic flow of Meyer type which satisfies physical boundary condition and whose sonic points are exceptional in a de Laval nozzle. For such a flow, its sonic curve must be located at the throat of the nozzle and the governing equation is strongly degenerate in the sense that the sonic curve is a characteristic degenerate boundary in the subsonic-sonic region, while in the sonic-supersonic region all characteristics from sonic points coincide, which are the sonic curve and never approach the supersonic region. It is proved that there exists uniquely such a smooth transonic flow near the throat of the nozzle, whose acceleration is Lipschitz continuous, if the wall of the nozzle is sufficiently flat. The works are jointed with Professor Zhouping Xin.

18. Huang Rongli

题目：On the second boundary value problem for a class of fully nonlinear flows I

摘要：A class of fully nonlinear flows is considered with nonlinear Neumann type boundary condition. We can show that the convexity is preserved for solutions of the fully nonlinear parabolic equations and exhibit the long time existence and convergence of the flow. In particular, it shows that one can prescribe the boundary value problems for a family of minimal Lagrangian graphs in Euclidean and pseudo-Euclidean space.

19. Zhu Anqiang

题目：Differential Harnack inequalities for the backward heat equation with potential under the harmonic-Ricci flow

摘要：Differential Harnack is a useful tool in Geometric Analysis. In this talk, we will consider A Perelman type Harnack inequality for the conjugate heat equation under the harmonic-Ricci flow. We also consider the Differential Harnack inequalities for the positive solutions of the backward heat equation with potential under the harmonic- Ricci flow.

20. Han Yingbo

题目： On the $p$-pseudoharmonic map heat flow

摘要：In this paper, we consider the heat flow for $p$-pseudoharmonic maps from a closed Sasakian manifold $(M^{2n+1},J,\theta)$ into a compact Riemannian manifold $(N^{m},g)$. We prove global existence and asymptotic convergence of the solution for the $p$-pseudoharmonic map heat flow, provided that the sectional curvature of the target manifold $N$ is nonpositive. Moreover, without the curvature assumption on the target manifold, we obtain global existence and asymptotic convergence of pseudoharmonic map heat flow as well when the its initial $p$-energy is sufficiently small.