In this paper, we consider eigenvalues of the Dirichlet biharmonic operator on a bounded domain in a hyperbolic space. We obtain universal bounds on the （k ＋ 1）th eigenvalue in terms of the first kth eigenvalues in...In this paper, we consider eigenvalues of the Dirichlet biharmonic operator on a bounded domain in a hyperbolic space. We obtain universal bounds on the （k ＋ 1）th eigenvalue in terms of the first kth eigenvalues independent of the domains.展开更多
设 x:M→R^(n+1)是凸域ΩR^n 上的严格凸函数 x_(n+1)=f(x_1,…,x_n)定义的一个局部强凸超曲面.如果 f 是下面方程的解,则称 M 为α相对极值超曲面:△ρ=(2-nα)/2(‖▽ρ‖~2)/ρ,ρ:=(det((a^2f)/(ax_iax_j)))^(1/(n+2)).2007年,贾...设 x:M→R^(n+1)是凸域ΩR^n 上的严格凸函数 x_(n+1)=f(x_1,…,x_n)定义的一个局部强凸超曲面.如果 f 是下面方程的解,则称 M 为α相对极值超曲面:△ρ=(2-nα)/2(‖▽ρ‖~2)/ρ,ρ:=(det((a^2f)/(ax_iax_j)))^(1/(n+2)).2007年,贾和李证明了存在一个仅依赖于维数 n 的正常数 K(n),如果|α|≥K(n),那么欧氏完备的α相对极值超曲面是椭圆抛物面.本文中我们利用 Calabi 度量给出了这个定理的一个简单证明.展开更多
In this paper, we discuss the Lagrangian angle and the K?hler angle of immersed surfaces in C2. Firstly, we provide an extension of Lagrangian angle, Maslov form and Maslov class to more general surfaces in C2 than La...In this paper, we discuss the Lagrangian angle and the K?hler angle of immersed surfaces in C2. Firstly, we provide an extension of Lagrangian angle, Maslov form and Maslov class to more general surfaces in C2 than Lagrangian surfaces, and then naturally extend a theorem by J.-M. Morvan to surfaces of constant K?hler angle, together with an application showing that the Maslov class of a compact self-shrinker surface with constant K?hler angle is generally non-vanishing. Secondly, we obtain two pinching results for the K?hler angle which imply rigidity theorems of self-shrinkers with K?hler angle under the condition that ∫M|h|2e-|x|2/2 dVM< ∞, where h and x denote, respectively, the second fundamental form and the position vector of the surface.展开更多
基金Supported by NSFC of China (No.10671181)Project of Henan Provincialdepart ment of Sciences and Technology (No.092300410143)NSF of Henan Provincial Education depart ment (No .2009A110010)
基金Supported by National Natural Science Foundation of China (1100107610926171+4 种基金10926172)Project of Henan Provincial Department of Sciences and Technology (092300410143)NSF of Henan Provincial Education Department (2009A1100102010A110008)Foundation of Young Backbone Teachers of Henan Normal University
基金supported by NSFC (11001076)Project of Henan Provincial department of Sciences and Technology (092300410143)+1 种基金NSF of Henan Provincial Education Department (2009A110010 2010A110008)
文摘In this paper, we consider eigenvalues of the Dirichlet biharmonic operator on a bounded domain in a hyperbolic space. We obtain universal bounds on the （k ＋ 1）th eigenvalue in terms of the first kth eigenvalues independent of the domains.
文摘设 x:M→R^(n+1)是凸域ΩR^n 上的严格凸函数 x_(n+1)=f(x_1,…,x_n)定义的一个局部强凸超曲面.如果 f 是下面方程的解,则称 M 为α相对极值超曲面:△ρ=(2-nα)/2(‖▽ρ‖~2)/ρ,ρ:=(det((a^2f)/(ax_iax_j)))^(1/(n+2)).2007年,贾和李证明了存在一个仅依赖于维数 n 的正常数 K(n),如果|α|≥K(n),那么欧氏完备的α相对极值超曲面是椭圆抛物面.本文中我们利用 Calabi 度量给出了这个定理的一个简单证明.
基金supported by National Natural Science Foundation of China(11671121,11871197)
文摘In this paper, we discuss the Lagrangian angle and the K?hler angle of immersed surfaces in C2. Firstly, we provide an extension of Lagrangian angle, Maslov form and Maslov class to more general surfaces in C2 than Lagrangian surfaces, and then naturally extend a theorem by J.-M. Morvan to surfaces of constant K?hler angle, together with an application showing that the Maslov class of a compact self-shrinker surface with constant K?hler angle is generally non-vanishing. Secondly, we obtain two pinching results for the K?hler angle which imply rigidity theorems of self-shrinkers with K?hler angle under the condition that ∫M|h|2e-|x|2/2 dVM< ∞, where h and x denote, respectively, the second fundamental form and the position vector of the surface.
