Lorentz-Minkowski空间中的乘积空间R~k×H~(n-k)

本文的主要结果是:Lorentz-Minkowski空间中介于两个同心伪圆柱面之间的完备、类空、常平均曲率超曲面必为伪圆柱面,即乘积空间R~k×H~(n-k)．对于常高阶平均曲率的情况,如果截曲率有下界,那么介于两个同心伪球面之间的完备类空超曲面必为伪球面．     

Real hypersurfaces in complex hyperbolic two‐plane Grassmannians with parallel Ricci tensor

In this paper we prove that there does not exist any Hopf real hypersurface in complex hyperbolic two‐plane Grassmannians with parallel Ricci tensor.     

Möbius Homogeneous Hypersurfaces with Three Distinct Principal Curvatures in Sn+1

Let x : M~n→ S~(n+1) be an immersed hypersurface in the(n + 1)-dimensional sphere S~(n+1). If, for any points p, q ∈ Mn, there exists a Mbius transformation φ :S~(n+1)→ S~(n+1) such that φox(Mn~) = x(M~n) and φ ox(p) = x(q), then the hypersurface is called a Mbius homogeneous hypersurface. In this paper, the Mbius homogeneous hypersurfaces with three distinct principal curvatures are classified completely up to a Mbius transformation.     

Remarks on biharmonic hypersurfaces in space forms

We consider closed biharmonic hypersurfaces in a Euclidean sphere and prove a rigidity result under a suitable condition on the scalar curvature. Moreover, we establish an integral formula involving the position vector for biharmonic hypersurfaces in space forms. As an application of this formula, we reobtain a result concerning the closed biharmonic hypersurfaces in Euclidean spheres that lie in a closed hemisphere.     

On the tangential Cauchy‐Fueter operators on nondegenerate quadratic hypersurfaces in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb {H}^2}$\end{document}

On quadratic hypersurfaces in $\mathbb {H}^2$, we find the explicit forms of tangential Cauchy‐Fueter operators and associated tangential Laplacians □_b. Then by using the Fourier transformation on the associated nilpotent Lie groups of step two, we construct the relative fundamental solutions to the tangential Laplacians and Szegö kernels on the nondegenerate quadratic hypersurfaces. It is different from the complex case that the quaternionic tangential structures on the nondegenerate quadratic hypersurfaces in $\mathbb {H}^2$ cannot be reduced to one standard model and the non‐homogeneous tangential Cauchy‐Fueter equations are solvable even in many convex cases.     

An extension of Barta’s Theorem and geometric applications

We prove an extension of a theorem of Barta and we give some geometric applications. We extend Cheng’s lower eigenvalue estimates of normal geodesic balls. We generalize Cheng-Li-Yau eigenvalue estimates of minimal submanifolds of the space forms. We show that the spectrum of the Nadirashvili bounded minimal surfaces in have positive lower bounds. We prove a stability theorem for minimal hypersurfaces of the Euclidean space, giving a converse statement of a result of Schoen. Finally we prove generalization of a result of Kazdan–Kramer about existence of solutions of certain quasi-linear elliptic equations. Bessa and Montenegro were partially supported by CNPq Grant.     

Complete Spacelike Submanifolds in de Sitter Space with Parallel Mean Curvature Vector Satisfying H2 = 4(n − 1)/n2

Recently, Montiel [7] proved that an n-dimensional (n 3) complete spacelike hypersurface in de Sitter space S^{1 1 +1}(1) with constant mean curvature H satisfying H² = 4 (n – 1)/n² which is not connected at infinity must be, up to rigidity motion, a certain hyperbolic cylinder. In this paper, we prove that Montiel's result still holds for higher codimensional spacelike submanifolds in de Sitter space Sⁿ _p ^+p(1).     

COMPLETE HYPERSURFACES WITH CONSTANT SCALAR CURVATURE IN A SPECIAL KIND OF LOCALLY SYMMETRIC MANIFOLD

In this paper, we investigate n-dimensional complete and orientable hypersufaces M n (n ≥ 3) with constant normalized scalar curvature in a locally symmetric manifold. Two rigidity theorems are obtained for these hypersurfaces.     

ON A CONJECTURE OF K. OGIUE FOR KAEHLER HYPERSURFACES

ＯＮ Ａ ＣＯＮＪＥＣＴＵＲＥ ＯＦ Ｋ． ＯＧＩＵＥ ＦＯＲ ＫＡＥＨＬＥＲ ＨＹＰＥＲＳＵＲＦＡＣＥＳ￥ＳＨＡＮＧＷＥＩＭＩＮＧＡｂｓｔｒａｃｔ：ＡｎａｆｆｉｒｍａｔｉｖｅａｎｓｗｅｒｔｏａｃｏｎｊｅｃｔｕｒｅｏｆＫ．Ｏｇｉｕｅｆｏｒｍｕｌａｔｅ...     

Different definitions of homogeneity of real hypersurfaces in ℂ2

The coincidence of two definitions of local homogeneity for real-analytic hypersurfaces in two-dimensional complex spaces is proved. It is shown that if any two germs of a Levi nondegenerate nonspherical surfaceM are equivalent, then this surface has a local Lie group structure:M then acts transitively on itself by left shifts, and each such shift is a local holomorphic transformation of ².Translated fromMatematicheskie Zametki, Vol. 64, No. 6, pp. 881–887, December, 1998.This paper was conceived and largely written during author's visits to the Universities of West Ontario and Regina (Canada) under the kind invitation of Professors B. Gilligan and A. Bowen. The author wishes to express his great indebtedness to them for providing excellent conditions for work and gratitude for useful discussions.