常曲率平面上的逆Bonnesen型不等式

利用积分几何中估计包含测度的思想给出常曲率平面上一些新的逆Bonnesen型不等式.这些不等式在欧氏平面上为著名的Bottema不等式的改进形式与新的逆Bonnesen型不等式.     

Geodesic Graphs with Constant Mean Curvature in Spheres

We study the existence and unicity of graphs with constant mean curvature in the Euclidean sphere of radius a. Given a compact domain , with some conditions, contained in a totally geodesic sphere S of and a real differentiable function on , we define the graph of in considering the height (x) on the minimizing geodesic joining the point x of to a fixed pole of . For a real number H such that |H| is bounded for a constant depending on the mean curvature of the boundary of and on a fixed number in (0,1), we prove that there exists a unique graph with constant mean curvature H and with boundary , whenever the diameter of is smaller than a constant depending on . If we have conditions on , that is, let be a graph over of a function, if |H| is bounded for a constant depending only on the mean curvature of and if the diameter of is smaller than a constant depending on H and , then we prove that there exists a unique graphs with mean curvature H and boundary . The existence of such a graphs is equivalent to assure the existence of the solution of a Dirichlet problem envolving a nonlinear elliptic operator.     

On Some Special Surfaces Connected with Convex Surfaces of the Lobachevskii Space

With a closed convex surface in a Lobachevskii space we associate four special surfaces: the inscribed and circumscribed spheres, a sphere rolling freely over the inner side of , and an equidistant surface over whose inner side rolls freely. We find an exact dependence between these four special surfaces.     

拟常曲率空间中具有常平均曲率的完备超曲面

主要研究了拟常曲率空间中具有常平均曲率的完备超曲面,得到了这类超曲面全脐的一个结果.即若Nn+1的生成元η∈TM,且a-2|b|=c(常数)>0,则当S<2 n-1~(1/2)(a-2|b|)时,M为全脐超曲面.     

常曲率空间中具正Ricci曲率的子流形

设S~(n+p)(1)是一单位球面,M~n是浸入S~(n+p)(1)的具有非零平行平均曲率向量的n维紧致子流形.证明了当n≥4,p≥2时,如果M~n的Ricci曲率不小于(n-2)(1+H~2),则M~n是全脐的或者M~n的Ricci曲率等于(n-2)(1+H~2),进而M~n的几何分类被完全给出.     

Bifurcation of Constant Mean Curvature Tori in Euclidean Spheres

We use equivariant bifurcation theory to show the existence of infinite sequences of isometric embeddings of tori with constant mean curvature (CMC) in Euclidean spheres that are not isometrically congruent to the CMC Clifford tori, and accumulating at some CMC Clifford torus.     

Coulomb Problem in a One-Dimensional Space with Constant Positive Curvature

We construct a complex transformation generalizing the known Hurwitz transformation in the Euclidean space for one-dimensional quantum mechanics. As in the case of the flat space, this transformation allows establishing the connection between the Coulomb problem and the oscillator problem with the Calogero–Sutherland potential added. We fully describe the motion in a Coulomb field in S ₁ and determine the energy spectrum and the wave functions with the correct normalizing constant.     

复射影空间中具有常数量曲率的实超曲面

将球面上常数量曲率超曲面推广到复射影空间中,得到此类实超曲面的某些积分不等式.     

Hypersurfaces with Constant Mean Curvature in Space Forms

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空间形式中常数量率的子流形

本文利用一个类似于Cheng和Yau引进的微分算子的新微分算子，得到了单位球面中常数量率的紧致子流形的一个刚性结果．     

关于凸体的Lp-曲率映象的不等式

Lutwak提出了凸体的Lp-曲率映象的概念,并证明了凸体与其Lp-曲率映象的体积之间的一个不等式.本文给出了Lutwak结果的一个一般形式,继而证明了凸体与其Lp-曲率映象的极的体积之间的一个不等式,并得到了凸体的Lp-投影体和Lp-曲率映象的体积之间的一个不等式.     

Radial Graphs with Constant Mean Curvature in the Hyperbolic Space

The existence is proved of radial graphs with constant mean curvature in the hyperbolic space H ⁿ⁺¹ defined over domains in geodesic spheres of H ⁿ⁺¹ whose boundary has positive mean curvature with respect to the inward orientation.     

二维严格凸赋范空间单位球面间等距映射的线性延拓

主要研究二维严格凸实赋范空间E和F的单位球面S_1(E)和S_1(F)之间的等距映射的线性延拓问题.利用二维严格凸赋范空间单位球面的性质得到:若等距映射V_0:S_1(E)→S_1(F)满足一定条件,则V_0可延拓为全空间E上的线性等距映射V:E→F.     

关于凸体的Lp-曲率映象的不等式

Lutwak提出了凸体的Lp-曲率映象的概念,并证明了凸体与其Lp-曲率映象的体积之间的一个不等式．本文给出了Lutwak结果的一个一般形式,继而证明了凸体与其Lp-曲率映象的极的体积之间的一个不等式,并得到了凸体的Lp-投影体和Lp-曲率映象的体积之间的一个不等式．     

常曲率空间中共球有限点集的一类几何不等式

本文首先证明了一个代数不等式,其次利用它得到了n维常曲率空间中共球有限点集的一类几何不等式,其特殊情况得到了文[3]中的主要结论.     

Lorentz空间型中具有常数量曲率类空超曲面

研究了Lorentz空间型中具有常数量曲率的紧致类空超曲面,得到了这类超曲面的内蕴刚性定理.     

Graphs of Constant Mean Curvature in Hyperbolic Space

We study the problem of finding constant mean curvature graphsover a domain of a totally geodesic hyperplane andan equidistant hypersurface Q of hyperbolic space. We findthe existence of graphs of constant mean curvature H overmean convex domains Q and with boundary for –H < H |h|, where H > 0 is the mean curvature of the boundary . Here h is the mean curvature respectively of the geodesic hyperplane (h= 0) and of the equidistant hypersurface (0 < |h|< 1). The lower bound on H is optimal.