Centroaffine Bernstein problems

C.P. Wang [Geom. Dedicata 51 (1994) 63–74] studied the Euler–Lagrange equation for the centroaffine area functional of hypersurfaces. In terms of a local representation of the hypersurface as a graph, this equation is a complicated, strongly non-linear fourth order PDE. We consider classes of solutions satisfying these equations together with completeness conditions. We also formulate appropriate centroaffine Bernstein problems and give partial solutions.     

Low codimensional submanifolds of Euclidean space with nonnegative isotropic curvature

In this paper we study compact submanifolds of Euclidean space with nonnegative isotropic curvature and low codimension. We determine their homology completely in the case of hypersurfaces and for some low codimensional conformally flat immersions.

     

Hypersurfaces with up to Six Double Points

Abstract

We present exponential formulas for the number of hypersurfaces of any dimension in a k-dimensional family displaying kordinary double points for k ≤ 6, extending our earlier work on surfaces.     

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).     

球面中等参超曲面及其焦流形的有限性定理

球面中等参超曲面理论近期获得蓬勃发展,它们在等距意义下的分类至今未完全解决,将证明球面中等参超曲面及其焦流形的微分同胚型有限.     

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].     

On biconservative hypersurfaces in ‐dimensional Riemannian space forms

In this paper, we study biconservative hypersurfaces in and . Further, we obtain complete explicit classifications of biconservative hypersurfaces in 4‐dimensional Riemannian space forms with exactly three distinct principal curvatures.     

S~4中等参超曲面的谱刻画

本文证明了如果Ｓ４中的具常平均曲率ｈ的超曲面Ｍ与其具平均曲率ｈ的等参超曲面Ｍ０（强）等谱，则Ｍ＝Ｍ０．     

Embedded Hypersurfaces in Sn+1 with Constant Mean Curvature and Spherical Boundary

It is proved that an embedded hypersurface in a hemisphere of the Euclidean unit spherewith constant mean curvature and spherical boundary inherits, under certainconditions, the symmetries of its boundary. In particular, spherical caps are theonly such hypersurfaces whose boundary are geodesic spheres.