A global pinching theorem for compact surfaces inS 3 with constant mean curvature

LetM be a compact minimal surface inS ³. Y. J. Hsu^[5] proved that if S₂22, thenM is either the equatorial sphere or the Clifford torus, whereS is the square of the length of the second fundamental form ofM, ·₂ denotes theL ²-norm onM. In this paper, we generalize Hsu's result to any compact surfaces inS ³ with constant mean curvature.Supported by NSFH.     

局部对称流形中具有常平均曲率的完备超曲面

讨论了局部对称黎曼流形中具有常平均曲率的完备超曲面的性质,通过Laplace算子的计算,得到一个关于第二基本形式模长平方S的拼挤定理,推广了已有的结果．     

de Sitter空间中具有常数量曲率的类空超曲面

研究了de Sitter空间中具有常数量曲率的类空超曲面,得到了曲面Mn关于截面曲率的一个刚性定理,并且额外获得关于de Sitter空间子流形的一个结论.     

A differentiable sphere theorem with positive Ricci curvature and reverse volume pinching

Let Mn be a compact, simply connected n (≥3)-dimensional Riemannian manifold without bound-ary and Sn be the unit sphere Euclidean space Rn+1. We derive a differentiable sphere theorem whenever themanifold concerned satisfies that the sectional curvature KM is not larger than 1, while Ric(M)≥n+2 4 and the volume V (M) is not larger than (1 + η)V (Sn) for some positive number η depending only on n.     

The second pinching theorem for hypersurfaces with constant mean curvature in a sphere

We generalize the second pinching theorem for minimal hypersurfaces in a sphere due to Peng–Terng, Wei–Xu, Zhang, and Ding–Xin to the case of hypersurfaces with small constant mean curvature. Let $M^n$ be a compact hypersurface with constant mean curvature $H$ in $S^{n+1}$ . Denote by $S$ the squared norm of the second fundamental form of $M$ . We prove that there exist two positive constants $\gamma (n)$ and $\delta (n)$ depending only on $n$ such that if $|H|\le \gamma (n)$ and $\beta (n,H)\le S\le \beta (n,H)+\delta (n)$ , then $S\equiv \beta (n,H)$ and $M$ is one of the following cases: (i) $S^{k}\Big (\sqrt{\frac{k}{n}}\Big )\times S^{n-k}\Big (\sqrt{\frac{n-k}{n}}\Big )$ , $\,1\le k\le n-1$ ; (ii) $S^{1}\Big (\frac{1}{\sqrt{1+\mu ^2}}\Big )\times S^{n-1}\Big (\frac{\mu }{\sqrt{1+\mu ^2}}\Big )$ . Here $\beta (n,H)=n+\frac{n^3}{2(n-1)}H^2+\frac{n(n-2)}{2(n-1)} \sqrt{n^2H^4+4(n-1)H^2}$ and $\mu =\frac{n|H|+\sqrt{n^2H^2+ 4(n-1)}}{2}$ .     

de sitter空间中具平行中曲率的完备类空子流形

本文推广了文[1]的结论，证明了de sitter空间Sp^n＋p（c）中具有平行中曲率的n维完备类空子流形的一个刚性定理．     

Hypersurfaces with constant scalar curvature and constant mean curvature

Non-spherical hypersurfaces inE ⁴ with non-zero constant mean curvature and constant scalar curvature are the only hypersurfaces possessing the following property: Its position vector can be written as a sum of two non-constant maps, which are eigenmaps of the Laplacian operator with corresponding eigenvalues the zero and a non-zero constant.     

A gap theorem for hypersurfaces of the sphere with constant scalar curvature one

We consider closed hypersurfaces of the sphere with scalar curvature one, prove a gap theorem for a modified second fundamental form and determine the hypersurfaces that are at the end points of the gap. As an application we characterize the closed, two-sided index one hypersurfaces with scalar curvature one in the real projective space. Received: October 12, 2001     

On the nondegeneracy of constant mean curvature surfaces

We prove that many complete, noncompact, constant mean curvature (CMC) surfaces are nondegenerate; that is, the Jacobi operator Δ_f + | A_f |² has no L² kernel. In fact, if ∑ has genus zero with k ends, and if f (∑) is embedded (or Alexandrov immersed) in a half-space, then we find an explicit upper bound for the dimension of the L² kernel in terms of the number of non-cylindrical ends. Our main tool is a conjugation operation on Jacobi fields which linearizes the conjugate cousin construction. Consequences include partial regularity for CMC moduli space, a larger class of CMC surfaces to use in gluing constructions, and a surprising characterization of CMC surfaces via spinning spheres. R.K. partially supported by NSF grants DMS-0076085 at GANG/UMass and DMS-9810361 at MSRI, and by a FUNCAP grant in Fortaleza, Brazil. J.R. partially supported by an NSF VIGRE grant at Utah. Received: January 2005; Accepted: June 2005     

A Bernstein theorem for complete spacelike hypersurfaces of constant mean curvature in Minkowski space

In this paper we prove a general Bernstein theorem on the complete spacelike constant mean curvature hypersurfaces in Minkowski space. The result generalizes the previous result of Cao-Shen-Zhu (1998) and Xin (1991). The proof again uses the fact that the Gauss map of a constant mean curvature hypersurface is harmonic, which was proved by K. T. Milnor (1983), and the maximum principle of S. T. Yau (1975).

     

Estimate for evolutionary surfaces of prescribed mean curvature and the convergence

In the paper, we will discuss the gradient estimate for the evolutionary surfaces of prescribed mean curvature with Neumann boundary value under the condition $f_\tau\ge -\kappa$, which is the same as the one in the interior estimate by K. Ecker and generalizes the condition $f_\tau\ge 0$ studied by Gerhardt etc. Also, based on the elliptic result obtained recently, we will show the longtime behavior of surfaces moving by the velocity being equal to the mean curvature.     

Constant mean curvature surfaces and mean curvature flow with non-zero Neumann boundary conditions on strictly convex domains

In this paper, we study nonparametric surfaces over strictly convex bounded domains in ${\mathbb{R}}^n$, which are evolving by the mean curvature flow with Neumann boundary value. We prove that solutions converge to the ones moving only by translation. And we will prove the existence and uniqueness of the constant mean curvature equation with Neumann boundary value on strictly convex bounded domains.     

Real hypersurfaces of with non-negative Ricci curvature

We prove the non-existence of Levi flat compact real hypersurfaces without boundary in , with non-negative totally real Ricci curvature.

     

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）.     

Sub-Riemannian constant mean curvature surfaces in the Heisenberg group as limits

The 3-dimensional Heisenberg group H together with its standard sub-Riemannian metric g₀ is viewed as the limit of a family of Riemannian manifolds, (H,g_u), u>0. For each u>0, we consider some invariant surfaces with constant mean curvature in (H,g_u). These surfaces of (H,g_u) have very nice limits as u0. We then define the mean curvature of a hypersurface in (H,g₀) to be the limit of its mean curvature in (H,g_u). We show that in a more general case, this definition is appropriate. Mathematics Subject Classification (2000) Principal 53C17; Secondary 22E25