The Volume of Hyperbolic Alternating Link Complements

If a hyperbolic link has a prime alternating diagram D, thenwe show that the link complement's volume can be estimated directlyfrom D. We define a very elementary invariant of the diagramD, its twist number t(D), and show that the volume lies betweenv₃(t(D) – 2)/2 and v₃(10t(D) – 10), where v₃ isthe volume of a regular hyperbolic ideal 3-simplex. As a consequence,the set of all hyperbolic alternating and augmented alternatinglink complements is a closed subset of the space of all completefinite-volume hyperbolic 3-manifolds, in the geometric topology.2000 Mathematics Subject Classification 57M25, 57N10.     

Hyperbolic ends with particles and grafting on singular surfaces

We prove that any 3-dimensional hyperbolic end with particles (cone singularities along infinite curves of angles less than π) admits a unique foliation by constant Gauss curvature surfaces. Using a form of duality between hyperbolic ends with particles and convex globally hyperbolic maximal (GHM) de Sitter spacetime with particles, it follows that any 3-dimensional convex GHM de Sitter spacetime with particles also admits a unique foliation by constant Gauss curvature surfaces. We prove that the grafting map from the product of Teichmüller space with the space of measured laminations to the space of complex projective structures is a homeomorphism for surfaces with cone singularities of angles less than π, as well as an analogue when grafting is replaced by “smooth grafting”.     

3维双曲空间中曲面的双曲Gauss映照和法Gauss映照

本文导出了3维双曲空间中曲面的双曲Gauss映照和法Gauss映照的关系,发现了一般的曲面由双曲Gauss映照和平均曲率函数唯一确定,并证明了双曲Gauss映照所满足的二阶线性椭圆方程,给出了两种形式的关于双曲Gauss映照的三阶非线性偏微分方程(组)的一个解.     

An orthogonal spline collocation alternating direction implicit method for second-order hyperbolic problems

On a rectangular region, we consider a linear second-order hyperbolicinitial-boundary value problem involving a mixed derivativeterm, continuous variable coefficients and non-homogeneous Dirichletboundary conditions. In comparison to the alternating directionimplicit Laplace-modified method of Fernandes (1997), we formulateand analyse a new parameter-free alternating direction implicitscheme in which the standard central difference formula is usedfor the time approximation and orthogonal spline collocationis used for the spatial discretization. We establish unconditionalstability of the scheme, and its optimal order in the discretemaximum norm in time and the H¹ norm in space. Numerical experimentsindicate that the new scheme, which has the same order as themethod of Fernandes (1997, Numer. Math., 77, 223–241),is more accurate. We also show that the new scheme is easilygeneralized to the second-order hyperbolic problems on rectangularpolygons. Extensions of the scheme to problems with discontinuouscoefficients, nonlinear problems, and problems with other boundaryconditions are also discussed.     

Zero and Infinite Curvature Limits of Hyperbolic Monopoles

We show that the zero curvature limit of the space of hyperbolicmonopoles gives the Euclidean monopoles, settling a conjectureof Atiyah. We also study the infinite curvature limit of thespace of hyperbolic monopoles, and show that the associatedrational maps appear explicitly here. 1991 Mathematics SubjectClassification 81T13, 53C07.     

四维Minkowski空间中类空超曲面的双曲高斯映射的奇点

介绍了四维Minkowski空间中类空超曲面的局部理论,定义了类空超曲面上的双曲高斯映射,双曲高度函数及距离平方函数,给出了一些定理的详细证明.介绍了一种证明高度函数是Morse族的新方法并应用Arnold等建立的Lagrange奇点理论对类空超曲面的双曲高斯映射的奇点进行了分类.     

The Normal Holonomy Group of Kahler Submanifolds

We study the (restricted) holonomy group Hol() of the normalconnection (shortened to normal holonomy group) of a Kählersubmanifold of a complex space form. We prove that if the normalholonomy group acts irreducibly on the normal space then itis linear isomorphic to the holonomy group of an irreducibleHermitian symmetric space. In particular, it is a compact groupand the complex structure J belongs to its Lie algebra. We prove that the normal holonomy group acts irreducibly ifthe submanifold is full (that is, it is not contained in a totallygeodesic proper Kähler submanifold) and the second fundamentalform at some point has no kernel. For example, a Kähler–Einsteinsubmanifold of CPⁿ has this property. We define a new invariant µ of a Kähler submanifoldof a complex space form. For non-full submanifolds, the invariantµ measures the deviation of J from belonging to the normalholonomy algebra. For a Kähler–Einstein submanifold,the invariant µ is a rational function of the Einsteinconstant. By using the invariant µ, we prove that thenormal holonomy group of a not necessarily full Kähler–Einsteinsubmanifold of CPⁿ is compact, and we give a list of possibleholonomy groups. The approach is based on a definition of the holonomy algebrahol(P) of an arbitrary curvature tensor field P on a vectorbundle with a connection and on a De Rham type decompositiontheorem for hol(P). 2000 Mathematics Subject Classification53C40 (primary), 53B25 (secondary).     

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.     

On Submanifolds with 2-Type Pseudo-Hyperbolic Gauss Map in Pseudo-Hyperbolic Space

In this paper, we study pseudo-Riemannian submanifolds of a pseudo-hyperbolic space \(\mathbb H^{m-1}_s (-1) \subset \mathbb E^m_{s+1}\) with 2-type pseudo-hyperbolic Gauss map. We give a characterization of proper pseudo-Riemannian hypersurfaces in \(\mathbb H^{n+1}_s (-1) \subset \mathbb E^{n+2}_{s+1}\) with non-zero constant mean curvature and 2-type pseudo-hyperbolic Gauss map. For \(n=2\), we prove classification theorems. In addition, we show that the hyperbolic Veronese surface is the only maximal surface fully lying in \(\mathbb H^4_2 (-1) \subset \mathbb H^{m-1}_2 (-1)\) with 2-type pseudo-hyperbolic Gauss map. Moreover, we prove that a flat totally umbilical pseudo-Riemannian hypersurface \(M^n_t\) of the pseudo-hyperbolic space \(\mathbb {H}^{n+1}_t(-1) \subset \mathbb E^{n+2}_{t+1}\) has biharmonic pseudo-hyperbolic Gauss map.     

The lightlike flat geometry on spacelike submanifolds of codimension two in Minkowski space

We introduce the notion of the lightcone Gauss–Kronecker curvature for a spacelike submanifold of codimension two in Minkowski space, which is a generalization of the ordinary notion of Gauss curvature of hypersurfaces in Euclidean space. In the local sense, this curvature describes the contact of such submanifolds with lightlike hyperplanes. We study geometric properties of such curvatures and show a Gauss–Bonnet type theorem. As examples we have hypersurfaces in hyperbolic space, spacelike hypersurfaces in the lightcone and spacelike hypersurfaces in de Sitter space.     

A Weierstrass representation for linear Weingarten spacelike surfaces of maximal type in the Lorentz-Minkowski space

In this work we extend the Weierstrass representation for maximal spacelike surfaces in the 3-dimensional Lorentz-Minkowski space to spacelike surfaces whose mean curvature is proportional to its Gaussian curvature (linear Weingarten surfaces of maximal type). We use this representation in order to study the Gaussian curvature and the Gauss map of such surfaces when the immersion is complete, proving that the surface is a plane or the supremum of its Gaussian curvature is a negative constant and its Gauss map is a diffeomorphism onto the hyperbolic plane. Finally, we classify the rotation linear Weingarten surfaces of maximal type.     

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

We obtain a gradient estimate for the Gauss maps from complete spacelike constant mean curvature hypersurfaces in Minkowski space into the hyperbolic space. As an application, we prove a Bernstein theorem which says that if the image of the Gauss map is bounded from one side, then the spacelike constant mean curvature hypersurface must be linear. This result extends the previous theorems obtained by B. Palmer [Pa] and Y.L. Xin [Xin1] where they assume that the image of the Gauss map is bounded. We also prove a Bernstein theorem for spacelike complete surfaces with parallel mean curvature vector in four-dimensional spaces. Received July 4, 1997 / Accepted October 9, 1997     

A d’Alembert Formula for Hopf Hypersurfaces

A Hopf hypersurface in complex hyperbolic space ${\mathbb{C}{\rm H}^n}$ is one for which the complex structure applied to the normal vector is a principal direction at each point. In this paper, Hopf hypersurfaces for which the corresponding principal curvature is small (relative to ambient curvature) are studied by means of a generalized Gauss map into a product of spheres, and it is shown that the hypersurface may be recovered from the image of this map, via an explicit parametrization.     

Gauss maps of the Ricci-mean curvature flow

In this paper, we investigate the Gauss maps of a Ricci-mean curvature flow. A Ricci-mean curvature flow is a coupled equation of a mean curvature flow and a Ricci flow on the ambient manifold. Ruh and Vilms (Trans Am Math Soc 149: 569–573, 1970) proved that the Gauss map of a minimal submanifold in a Euclidean space is a harmonic map, and Wang (Math Res Lett 10(2–3):287–299, 2003) extended this result to a mean curvature flow in a Euclidean space by proving its Gauss maps satisfy the harmonic map heat flow equation. In this paper, we deduce the evolution equation for the Gauss maps of a Ricci-mean curvature flow, and as a direct corollary we prove that the Gauss maps of a Ricci-mean curvature flow satisfy the vertically harmonic map heat flow equation when the codimension of submanifolds is 1.     

A Petrov-Galerkin method with quadrature for elliptic boundary value problems

We propose and analyse a fully discrete Petrov–Galerkinmethod with quadrature, for solving second-order, variable coefficient,elliptic boundary value problems on rectangular domains. Inour scheme, the trial space consists of C² splines of degreer 3, the test space consists of C⁰ splines of degree r –2, and we use composite (r – 1)-point Gauss quadrature.We show existence and uniqueness of the approximate solutionand establish optimal order error bounds in H², H¹ and L² norms.     

WEIERSTRASS REPRESENTATION FORSURFACES WITH PRESCRIBED NORMALGAUSS MAP AND GAUSS CURVATURE IN H~3

The author obtains a Weierstrass representation for surfaces with prescribed normal Gauss map and Gauss curvature in H3. A differential equation about the hyperbolic Gauss map is also obtained, which characterizes the relation among the hyperbolic Gauss map, the normal Gauss map and Gauss curvature. The author discusses the harmonicity of the normal Gauss map and the hyperbolic Gauss map from surface with constant Gauss curvature in H3 to S2 with certain altered conformal metric. Finally, the author considers the surface whose normal Gauss map is conformal and derives a completely nonlinear differential equation of second order which graph must satisfy.     

Circle Actions and Morse Theory on Quaternion-Kahler Manifolds

In this paper we study quaternion-Kähler manifolds endowedwith an isometric S¹-action. We consider the corresponding momentmap µ and prove that the only compact quaternion-Kählermanifold with positive scalar curvature which admits an isometriccircle action free on µ^–1(0) is the quaternionicprojective space HPⁿ.     

Finsler Metrics of Constant Positive Curvature on the Lie Group S3

Guided by the Hopf fibration, a family (indexed by a positiveconstant K) of right invariant Riemannian metrics on the Liegroup S³ is singled out. Using the Yasuda–Shimada paperas an inspiration, a privileged right invariant Killing fieldof constant length is determined for each K > 1. Each suchRiemannian metric couples with the corresponding Killing fieldto produce a y-global and explicit Randers metric on S³. Employingthe machinery of spray curvature and Berwald's formula, it isproved directly that the said Randers metric has constant positiveflag curvature K, as predicted by Yasuda–Shimada. It isexplained why this family of Finslerian space forms is not projectivelyflat.     

Spectral transformation of constant mean curvature surfaces in H3 and Weierstrass representation

By using the method of integrable system, we study the deformation of constant mean curvature surfaces in three-dimensional hyperbolic space form H₃. We also obtain a Weierstrass representation formula of the constant mean curvature surfaces with mean curvature greater than 1     

Spacelike submanifolds of codimension two in anti-de Sitter space

In this paper, we study spacelike submanifolds of codimension two in anti-de Sitter space from the viewpoint of Legendrian singularity theory. We introduce the notion of the anti-de Sitter normalized Gauss map which is a generalization of the ordinary notion of Gauss map of hypersurfaces in Euclidean space. We also introduce the AdS-normalized Gauss–Kronecker curvature for a spacelike submanifold of codimention two in anti-de Sitter space. In the local sense, this curvature describes the contact of submanifolds with some model surfaces.