Singularity analysis of Lorentzian hypersurfaces on pseudo n‐spheres

This paper considers the one parameter families of extrinsic differential geometries of Lorentzian hypersurfaces on pseudo n‐spheres. We give a continuous relationship among the three types Gauss indicatrices by one parameter map. Meanwhile, we also give the singularity analysis of the one parameter Gauss indicatrices of Lorentzian hypersurfaces on pseudo n‐spheres by the Legendrian singularity theory. Copyright © 2014 John Wiley & Sons, Ltd.     

Immersions of Finite Geometric Type in Euclidean Spaces

In this paper, we introduce the class of hypersurfaces of finitegeometric type. They are defined as the ones that share the basicdifferential topological properties of minimal surfaces of finite totalcurvature. We extend to surfaces in this class the classical theorem ofOsserman on the number of omitted points of the Gauss mapping ofcomplete minimal surfaces of finite total curvature. We give aclassification of the even-dimensional catenoids as the only even-dimensional minimal hypersurfaces of R ⁿ of finite geometric type.     

Systolic Inequalities and Minimal Hypersurfaces

We give a short proof of the systolic inequality for the n-dimensional torus. The proof uses minimal hypersurfaces. It is based on the Schoen–Yau proof that an n-dimensional torus admits no metric of positive scalar curvature.     

New extension phenomena for solutions of tangential Cauchy–Riemann equations

In our recent work, we showed that C^∞ CR-diffeomorphisms of real-analytic Levi-nonflat hypersurfaces in ?² are not analytic in general. This result raised again the question on the nature of CR-maps of a real-analytic hypersurfaces.

In this paper, we give a complete picture of what CR-maps actually are. First, we discover an analytic continuation phenomenon for CR-diffeomorphisms which we call the sectorial analyticity property. It appears to be the optimal regularity property for CR-diffeomorphisms in general. We emphasize that such type of extension never appeared previously in the literature. Second, we introduce the class of Fuchsian type hypersurfaces and prove that (infinitesimal generators of) CR-automorphisms of a Fuchsian type hypersurface are still analytic. In particular, this solves a problem formulated earlier by Shafikov and the first author.

Finally, we prove a regularity result for formal CR-automorphisms of Fuchsian type hypersurfaces.     

A characterization of totally <Emphasis Type="Italic">η</Emphasis>-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form

We give a characterization of totally η-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form in terms of totally umbilical condition for the holomorphic distribution on real hypersurfaces. We prove that if the shape operator A of a real hypersurface M of a complex space form M ⁿ (c), c ≠ 0, n ⩾ 3, satisfies g(AX, Y) = ag(X, Y) for any X, Y ∈ T ₀(x), a being a function, where T ₀ is the holomorphic distribution on M, then M is a totally η-umbilical real hypersurface or locally congruent to a ruled real hypersurface. This condition for the shape operator is a generalization of the notion of η-umbilical real hypersurfaces.     

When does the Hessian determinant vanish identically?

In 1851, Hesse claimed that the Hessian determinant of a homogeneous polynomial f vanishes identically if and only if the projective hypersurface V (f) is a cone. We follow the lines of the 1876 paper of Gordan and Noether to give a proof of Hesses claim for curves and surfaces. For higher dimensional hypersurfaces, the claim is wrong in general. We review the construction of polynomials with vanishing Hessian determinant but V (f) not being a cone. For three dimensional hypersurfaces the latter gives, again, the complete answer to the question asked in the title.     

An algebraic characterization of holomorphic nondegeneracy for real algebraic hypersurfaces and its application to CR mappings

We give a new algebraic characterization of holomorphic nondegeneracy for embedded real algebraic hypersurfaces in , . We then use this criterion to prove the following result about real analyticity of smooth CR mappings: any smooth CR mapping H between a real analytic hypersurface and a rigid polynomial holomorphically nondegenerate hypersurface is real analytic, provided the map H is not totally degenerate in the sense of Baouendi and Rothschild. Received September 19, 1997     

Hopf hypersurfaces of small Hopf principal curvature in $${\mathbb{C}{\rm H}^2}$$

Using the methods of moving frames and exterior differential systems, we show that there exist Hopf hypersurfaces in complex hyperbolic space with any specified value of the Hopf principal curvature α less than or equal to the corresponding value for the horosphere. We give a construction for all such hypersurfaces in terms of Weierstrass-type data, and also obtain a classification of pseudo-Einstein hypersurfaces in .      

Cohomogeneity One Hypersurfaces of the Hyperbolic Space

In this paper we extend to the hyperbolic space results firstly obtainedin Podestà and Spiro (Ann. Global Anal. Geom. 13(2) (1995), 169–184) for compact cohomogeneity, Riemannian manifoldsimmersed as hypersurfaces of the Euclidean space and in Seixas(Ph.D. Thesis, 1996), wherethe complete case was studied. Both works give sufficient conditionsfor the hypersurface to be of revolution. We study cohomogeneity, complete hypersurfaces of the hyperbolic space and prove a similar resultof Seixas (Ph.D. Thesis, 1996), obtaining also a class of nonrotational examples.     

The structure vector field and structure Jacobi operator of real hypersurfaces in nonflat complex space forms

In this paper we determine the real hypersurfaces for which the structure Jacobi operator commutes over both the Ricci tensors and structure tensors (for a definition of the operator see Sect. 1). We prove that such hypersurfaces are homogneous real hypersurfaces of type (A) and are a special class of Hopf hypersurfaces.     

Dimca Hypersurfaces and Nagata Automorphisms

Choudary–Dimca [CD ] proved that Dimca hypersurfaces are diffeomorphic to ℝ²ⁿ and some of the hypersurfaces with dimension 3 are isomorphic to the plane and the embeddings are equivalent to linear embeddings. However, neither an isomorphism nor a trivializing polynomial automorphism was given explicitly. The purpose of the present paper is to give these explicitly.     

de Sitter空间S1m+1中具有平行Blaschke张量的正则类空超曲面

本文引入两个以de Sitter空间为模型的非齐性坐标来覆盖共形空间Q₁^m+1.利用球面S^m+1中超曲面的Möbius几何的方法,本文研究了Q₁^m+1中正则类空超曲面的共形几何.作为其结果,本文对所有具有平行Blaschke张量的正则类空超曲面进行了完全分类.     

Circular trajectories on real hypersurfaces in a nonflat complex space form

In this paper we study which trajectories for Sasakian magnetic fields are circles on certain standard real hypersurfaces which are called hypersurfaces of type A in a nonflat complex space form. We also give a characterization of these real hypersurfaces by such a circular property of trajectories for Sasakian magnetic fields.     

Diagonal F-Thresholds on Binomial Hypersurfaces

In this article, we study the diagonal F-threshold, that is, the F-threshold of the maximal ideal with respect to the maximal ideal itself. In particular, we give a formula of the diagonal F-thresholds for binomial hypersurfaces, and compare the diagonal F-thresholds with F-pure thresholds for strongly F-regular binomial hypersurfaces.

As a result, we prove that those diagonal F-thresholds always exist and they are rational numbers in that case.     

Integral inequalities for maximal spacelike hypersurfaces in the indefinite space form

We give two intrinsic integral inequalities for compact maximal spacelike hypersurfaces in the indefinite space form and a sufficient and necessary condition for such hypersurfaces to be totally geodesic.     

Surfaces in three-dimensional space forms with divergence-free stress-bienergy tensor

We introduce the notion of biconservative hypersurfaces, that is hypersurfaces with conservative stress-energy tensor with respect to the bienergy. We give the (local) classification of biconservative surfaces in three-dimensional space forms.     

Topological and metric rigidity teorems for hypersurfaces in a hyperbolic space

In this paper we study the topological and metric rigidity of hypersurfaces in ℍ ⁿ⁺¹, the (n + 1)-dimensional hyperbolic space of sectional curvature −1. We find conditions to ensure a complete connected oriented hypersurface in ℍ ⁿ⁺¹ to be diffeomorphic to a Euclidean sphere. We also give sufficient conditions for a complete connected oriented closed hypersurface with constant norm of the second fundamental form to be totally umbilic.     

Minimizing Constant Mean Curvature Hypersurfaces in Hyperbolic Space

We study the constant mean curvature (CMC) hypersurfaces in whose asymptotic boundaries are closed codimension-1 submanifolds in . We consider CMC hypersurfaces as generalizations of minimal hypersurfaces. We naturally generalize some notions of minimal hypersurfaces like being area-minimizing, convex hull property, exchange roundoff trick to CMC hypersurface context. We also give a generic uniqueness result for CMC hypersurfaces in hyperbolic space.