2020-2021第一学期几何讨论班

发布者:赖惠林发布时间:2020-09-01浏览次数:14

2020-2021第一学期几何讨论班

 

时间:2020.9.  

地点:理工南楼616      

主讲人:王丽莉    

题目:On the fundamental gap of spheres

 

时间:2020.9.  

地点:理工南楼616      

主讲人:林和子    

题目:Gradient estimate and Liouville theorems for p-harmonic maps

 

时间:2020.10.21

地点:理工南楼616    

主讲人:王孝振

题目:Moebius geometry of submanifolds in S^n

摘要:In this paper we define a Moebius invariant metric and a Moebius invariant second fundamental form for submanifolds in S^n and show that in case of a hypersurface with n  4 they determine the hypersurface up to Moebius transformations. Using these Moebius invariants we calculate the rst variation of the moebius volume functional. We show that any minimal surface in S^n is also Moebius minimal and that the image in S^n of any minimal surface in R^n unter the inverse of a stereographic projection is also Moebius minimal. Finally we use the relations between Moebius invariants to classify all surfaces in S^3 with vanishing Moebius form.

 

时间:2020.10.28 

地点:理工南楼616      

主讲人:王孝振    

题目:Moebius geometry of submanifolds in S^n-2

摘要:In this paper we define a Moebius invariant metric and a Moebius invariant second fundamental form for submanifolds in S^n and show that in case of a hypersurface with n  4 they determine the hypersurface up to Moebius transformations. Using these Moebius invariants we calculate the rst variation of the moebius volume functional. We show that any minimal surface in S^n is also Moebius minimal and that the image in S^n of any minimal surface in R^n unter the inverse of a stereographic projection is also Moebius minimal. Finally we use the relations between Moebius invariants to classify all surfaces in S^3 with vanishing Moebius form.

 

时间:2020.11.4 

地点:理工南楼616      

主讲人:林丽妙    

题目:A Moebius rigidity of compact Willmore hypersurfaces in S^{n+1}

摘要:Let x : M^n  S^n+1 be an immersed hypersurface without umbilical point, then one can define the Möbius metric g, the Möbius second fundamental form B and the Blaschke tensor A on the hypersurface M^n which are invariant under the Möbius transformation group of S^n+1. A hypersurface is called a Willmore hypersurface if it is the critical point of the volume functional of Mn with respect to the Möbius metric g. In this paper, we prove that if the hypersurface x is a compact Willmore hypersurface without umbilical point, then

the equality holds if and only if the hypersurface Mn is Möbius equivalent to one of the Willmore tori


where the tensor.

 

时间:2020.11.11

地点:理工南楼616       

主讲人:林丽妙    

题目:A Moebius rigidity of compact Willmore hypersurfaces in S^{n+1}

摘要:Let x : M^n  S^n+1 be an immersed hypersurface without umbilical point, then one can define the Möbius metric g, the Möbius second fundamental form B and the Blaschke tensor A on the hypersurface M^n which are invariant under the Möbius transformation group of S^n+1. A hypersurface is called a Willmore hypersurface if it is the critical point of the volume functional of Mn with respect to the Möbius metric g. In this paper, we prove that if the hypersurface x is a compact Willmore hypersurface without umbilical point, then


the equality holds if and only if the hypersurface Mn is Möbius equivalent to one of the Willmore tori


where the tensor.

 

时间:2020. 11.18  

地点:理工南楼616    

主讲人:王鹏   

题目:On Simons inequality and its applications.

 

时间:2020. 12.2  

地点:理工南楼616    

主讲人:林和子    

题目:The Isoperimetric Inequality for a Minimal Submanifold in Euclidean Space.

摘要:We prove an isoperimetric inequality which holds for minimal submanifolds in Euclidean space of arbitrary dimension and codimension. Our estimate is sharp if the codimension is at most 2.


时间:2020. 12.9 

地点:理工南楼616     

主讲人:王丽莉    

题目:An Estimate of the Gap of the First Two Eigenvalues in the Schrodinger Operator.

 

时间:2020. 12.23  

地点:理工南楼616    

主讲人:王丽莉    

题目:Estimates on the modulus of expansion for vector fields solving nonlinear equations.

摘要:In this article, by extending the method of Andrews and Clutterbuck (2011) [2] we prove a sharp estimate on the expansion modulus of the gradient of the logarithm of the parabolic kernel to the Schrödinger operator with convex potential on a bounded convex domain. The result improves an earlier work of BrascampLieb which asserts the log-concavity of the parabolic kernel. We also give an alternate proof to a corresponding estimate on the first eigenfunction of the Schrödinger operator, obtained firstly by Andrews and Clutterbuck via the study of the asymptotics to a parabolic problem. Our proof is more direct via an elliptic maximum principle. An alternate proof of the fundamental gap theorem of Andrews and Clutterbuck (2011) [2], by considering the quotient of moduli of continuity, is also obtained. Moreover we derive a Neumann eigenvalue comparison result and some other lower estimates on the first Neumann eigenvalue for Laplace operator with a drifting term, including an explicit estimate on a conjecture of P. Li.

 

时间:2020. 12.30  

地点:理工南楼616     

主讲人:王鹏    

题目:Willmore Stability of the Lawson minimal surfaces $\xi_{g,1}$.

摘要:The generalized Willmore conjecture, proposed by Rob Kusner,  states that the Lawson minimal surface $\xi_{g,1}$ minimizes Willmore energy among all closed surfaces of genus $g>1$.  So far there are very few progress on this conjecture. A natural idea is to consider the Willmore stability of them. In this talk we will show that they are strictly Willmore stable both in $S^3$ and in $S^n$ via $S^3\subset S^n$, based on a joint with Prof. Kusner.


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