Curvature Computations of 2-Manifolds in $\mathbb{R}^k$

In this paper, we provide simple and explicit formulas for computing Riemannian curvatures, mean curvature vectors, principal curvatures and principal directions for a 2-dimensional Riemannian manifold embedded in IR^k with k ≥ 3.     

中曲率大于零的非凸曲面

本文研究了曲面的曲率问题.利用积分几何方法和旋转曲面性质,构造出一个欧氏空间R3中紧致光滑的中曲率H大于零的非凸曲面,并得到了关于紧致光滑曲面曲率的几个不等式.     

三维欧氏空间中的一类Weingarten螺旋面

通过求解相关的非线性常微分方程,构造了三维欧氏空间中主曲率之差为常数的螺旋面,并证明这类曲面的广泛存在性.     

Dupin超曲面的主曲率的重数

作者证明了如下结果 :设M是空间形式的闭定向Dupin超曲面 ,其截面曲率为正 ,M至少有两个不同主曲率 .如果除最小 (或最大 )主曲率外 ,M的其余主曲率均为常数 ,则最小 (或最大 )主曲率的重数大于 2 .     

一类推广的中曲率水平集的运动

研究了根据一个光滑曲面的主曲率函数的经典运动短时存在及唯一性,推广了Evans和Spruck的结果。     

3维单位球面中保主曲率的曲面的变形

在[1]中,陈省身大师讨论了欧氏空间 E~3保主曲率的曲面的变形,本文,考虑了维单位球面 S~3中相同的问题,并给出了分类定理。     

Lorentz-Minkowski空间中给定主曲率函数的旋转超曲面

给出(n,1)型orentz-Minkowski空间中给定主曲率函数的旋转类空超曲面的位置向量场,通过计算超曲面的主曲率,证明了这类超曲面的存在性.     

R31中混合型曲面保主曲率等距变形

本文研究了Minkowski空间R31曲面的等距变形问题.建立了R31中曲面的共形、等距等概念.推广了O.Bonnet和S.S.Chern关于欧氏空间的结论.对R31出现的新情况--曲面的中曲率梯度类光作了一定探讨,得出的主要结果为:非平坦的、允许保主曲率等距变形的曲面一定不是W-曲面.     

复射影空间中曲率齐性实超曲面

本文讨论复射影空间中曲率齐性实超曲面，在适当条件下证明了它与复射影空间中等参超曲面等价，因此得到它的局部结构。     

BCKLUND TRANSFORMATION ON SURFACESWITH CONSTANT MEAN CURVATURE IN R~(2.1)

In this paper, the authors obtain the Backlund transformation on time-likesurfaces with constant mean curvature in R~(2,1). Using this transformation, families ofsurfaces with constant mean curvature from known ones can be constructed.     

Damped Oscillatory Integrals and the Boundedness Problem for Maximal Operators

Let S ⊂ ℝR ^{n +1} be a real-analytic hypersurface with surface measure dσ, and let ψ be a smooth nonnegative compactly supported cutoff function. Consider the surface measure dμ _q = ψ|Λ(X)|^q dσ, where Λ(X) is a damping factor determined by the matrices of the first and second fundamental forms of the surface. We show that its Fourier transform decays for large |ξ| as O (|ξ|^−(1/2+ε)), ε > 0, provided that q > 3/2. We also consider applications involving maximal operators associated with means of functions over hypersurfaces.__________Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 39, No. 2, pp. 70–74, 2005Original Russian Text Copyright © by I. A. Ikromov     

A Generalization of Intersection Formulae of Integral Geometry

We establish extensions of the Crofton formula and, under some restrictions, of the principal kinematic formula of integral geometry from curvature measures to generalized curvature measures of convex bodies. We also treat versions for finite unions of convex bodies. As a consequence, we get a new intuitive interpretation of the area measures of Aleksandrov and Fenchel–Jessen.     

On the topology of isoparametric hypersurfaces with four distinct principal curvatures

Let be the pair of multiplicities of an isoparametric hypersurface in the unit sphere with four distinct principal curvatures -w.r.g., we assume that . In the present paper we prove that, in the case 4B2 of U. Abresch (Math. Ann. 264 (1983), 283-302) (i.e., where ), must be either 2 or 4. As a by-product, we prove that the focal manifold of an isoparametric hypersurface is homeomorphic to a bundle over if one of the following conditions holds: (1) and or ; (2) and . This generalizes partial results of Wang (1988) about the topology of Clifford type examples. Consequently, the hypersurface is homeomorphic to an iterated sphere bundle under the above condition.

     

A characterization of round spheres

A new characterization of geodesic spheres in the simply connected space forms in terms of higher order mean curvatures is given: An immersion of an dimensional compact oriented manifold without boundary into dimensional Euclidean space, hyperbolic space or the open half sphere is a totally umbilic immersion if, for some the -th mean curvature does not vanish and the ratio is constant.

     

Complete surfaces in the hyperbolic space with a constant principal curvature

In this paper we study complete orientable surfaces with a constant principal curvature R in the 3‐dimensional hyperbolic space H ³. We prove that if R² > 1, such a surface is totally umbilical or umbilically free and it can be described in terms of a complete regular curve in H ³. When R² ≤ 1, we show that this result is not true any more by means of several examples. This contradicts a previous statement by Zhisheng [6]. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Evolution equation of the Gauss curvature under hypersurface flows and its applications

In this paper, we derive evolution equation of the integral of the Gauss curvature on an evolving hypersurface. As an application, we obtain a monotone quantity on the level surface of the potential function on a 3-dimensional steady gradient Ricci soliton with positive sectional curvature, and prove that such a soliton is rotationally symmetric outside of a compact set under a curvature decaying assumption. Along the way we will also apply our evolution equation to some other cases.     

Hypersurfaces with constant mean curvature in a hyperbolic space

Let Mⁿ be an n-dimensional complete connected and oriented hypersurface in a hyperbolic space Hⁿ⁺¹（c） with non-zero constant mean curvature H and two distinct principal curvatures. In this paper, we show that （1） if the multiplicities of the two distinct principal curvatures are greater than 1,then Mⁿ is isometric to the Riemannian product S^k（r）×H^n-k（-1/（r² + ρ²））, where r > 0 and 1 < k < n - 1;（2）if H² > -c and one of the two distinct principal curvatures is simple, then Mⁿ is isometric to the Riemannian product S^n-1（r） × H¹（-1/（r² +ρ²）） or S¹（r） × H^n-1（-1/（r² +ρ²））,r > 0, if one of the following conditions is satisfied （i） S≤（n-1）t²2+c²t^-22 on Mⁿ or （ii）S≥ （n-1）t²1+c²t^-21 on Mⁿ or（iii）（n-1）t²2+c²t^-22≤ S≤（n-1）t²1+c²t^-21 on Mⁿ, where t₁ and t₂ are the positive real roots of （1.5）.