Radial Graphs with Constant Mean Curvature in the Hyperbolic Space

The existence is proved of radial graphs with constant mean curvature in the hyperbolic space H ⁿ⁺¹ defined over domains in geodesic spheres of H ⁿ⁺¹ whose boundary has positive mean curvature with respect to the inward orientation.     

A Note on Octic Hypersurfaces with Many Nodes

In the present note we use rank-2-bundles over to construct octic hypersurfaces with many nodes. We give an example with 128 nodes.The author gratefully acknowledges the support of the DFG priority program ‘Global methods in complex geometry’.     

Generalized doubling constructions for constant mean curvature hypersurfaces in <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis>+1</Superscript>

The sphere S ⁿ⁺¹ contains a simple family of constant mean curvature (CMC) hypersurfaces of the form C _t : = S ^p (cos t) × S ^q (sin t) for p + q = n and called the generalized Clifford hypersurfaces. This paper demonstrates that new, topologically non-trivial CMC hypersurfaces resembling a pair of neighbouring generalized Clifford tori connected to each other by small catenoidal bridges at a sufficiently symmetric configuration of points can be constructed by perturbative PDE methods. That is, one can create an approximate solution by gluing a rescaled catenoid into the neighbourhood of each point; and then one can show that a perturbation of this approximate hypersurface exists, which satisfies the CMC condition. The results of this paper generalize those of the authors in [3].     

de Sitter空间中具有常均曲率的类空超曲面

设M是deSitter空间中具有常均曲率的类空超曲面,本文给出M是全脐的或是等参的一些条件．     

Finite-volume hyperbolic 4-manifolds that share a fundamental polyhedron

It is known that the volume function for hyperbolic manifolds of dimension 3 is finite-to-one. We show that the number of nonhomeomorphic hyperbolic 4-manifolds with the same volume can be made arbitrarily large. This is done by constructing a sequence of finite-sided finite-volume polyhedra with side-pairings that yield manifolds. In fact, we show that arbitrarily many nonhomeomorphic hyperbolic 4-manifolds may share a fundamental polyhedron. As a by-product of our examples, we also show in a constructive way that the set of volumes of hyperbolic 4-manifolds contains the set of even integral multiples of 4π²/3. This is “half” the set of possible values for volumes, which is the integral multiples of 4π²/3 due to the Gauss-Bonnet formula Vol(M) = 4π²/3 · χ(M).     

球面2型超曲面

本文在无先念条件下,证明了Ｃｈｅｎ［１］关于２型超曲面的一个定理．     

A NOTE ON THE QUADRILATERAL MESH CONDITION RDP（N,ψ）

The main aim of this paper is to show that the quadrilateral mesh condition RDP（N, ψ） is only sufficient but not necessary for the optimal order error estimate of the Q isoparametric element in the Hi norm.     

Ricci η‐recurrent real hypersurfaces in 2‐dimensional nonflat complex space forms

Let M be a Hopf hypersurface in a nonflat complex space form $M^2(c)$, $c\ne 0$, of complex dimension two. In this paper, we prove that M has η‐recurrent Ricci operator if and only if it is locally congruent to a homogeneous real hypersurface of type (A) or (B) or a non‐homogeneous real hypersurface with vanishing Hopf principal curvature. This is an extension of main results in [17, 21] for real hypersurfaces of dimension three. By means of this result, we give some new characterizations of Hopf hypersurfaces of type (A) and (B) which generalize those in [14, 18, 26].     

双曲空间Hn+1(-1)中完备超曲面

本文研究了双曲空间Hn+1(-1)中具有常数量曲率的完备超曲面.利用活动标架的方法,得到此类超曲面的两个刚性定理.     

Meshfree isoparametric point interpolation method (IPIM) for evaporative laser drilling

An isoparametric point interpolation method (IPIM) has been developed to analyze evaporative laser machining. The method is based on isoparametric representation of the unknown in the local domain. It also uses a simple strong form, but shows to be powerful enough to handle highly localized boundary conditions. Solution in a typical influence domain is approximated by a polynomial function. The problem is geometrically nonlinear because the domain is not known a priori due to material removal in machining. An iterative scheme is introduced to solve the nonlinear problem. The material removal is handled by redistributing points in the domain. This renders the point distribution non-uniform same as random distribution. Three different boundary conditions considered are of essential, convection, and laser irradiation type. The numerical results show very good agreement with those by FEM and BEM.