三维球面中的旋转二次曲面作为Weingarten曲面

空间形式中的管道线性 Weingarten 曲面

Channel linear Weingarten surfaces in space forms

本文在 Lie 球面几何背景下统一研究了空间形式中的管道线性 Weingarten 曲面，证明了其等温性与旋转曲面性质，并给出了常 Gauss 曲率管道曲面及相关线性 Weingarten 曲面的显式参数化。

On linear Weingarten surfaces

In this paper we study surfaces in Euclidean 3-space that satisfy a Weingarten condition of linear type as $κ_1=m κ_2 +n$, where $m$ and $n$ are real numbers and $κ_1$ and $κ_2$ denote the principal curvatures at each point of the surface. We investigate the possible existence of such surfaces parametrized by a uniparametric family of circles. Besides the surfaces of revolution, we prove that not exist more except the case $(m,n)=(-1,0)$, that is, if the surface is one of the classical examples of minimal surfaces discovered by Riemann.

A characterization of revolution quadrics by a system of partial differential equations

It is shown that existence of a global solution to a particular nonlinear system of second order partial differential equations on a complete connected Riemannian manifold has topological and geometric implications and that in the domain of positivity of such solution its reciprocal is the radial function of only one of the following rotationally symmetric hypersurfaces in $\Rn$: paraboloid, ellipsoid, one sheet of a two-sheeted hyperboloid, and a hyperplane.

脐斜率与三次Weingarten曲面

Umbilic slopes and cubic Weingarten surfaces

本文研究脐斜率并分类满足κ₂=cκ₁³的紧致浸入曲面，证明其每个连通分量均为旋转椭球，方法涉及齐次Hessian恒等式、唯一延拓及四阶增长估计。

Rotational symmetry of Weingarten spheres in homogeneous three-manifolds

Let $M$ be a simply connected homogeneous three-manifold with isometry group of dimension $4$, and let $Σ$ be any compact surface of genus zero immersed in $M$ whose mean, extrinsic and Gauss curvatures satisfy a smooth elliptic relation $Φ(H,K_e,K)=0$. In this paper we prove that $Σ$ is a sphere of revolution, provided that the unique inextendible rotational surface $S$ in $M$ that satisfies this equation and touches its rotation axis orthogonally has bounded second fundamental form. In particular, we prove that: (i) any elliptic Weingarten sphere immersed in $\mathbb{H}^2\times \mathbb{R}$ i…

The Bi-normal fields on spacelike surfaces in $\mathbb R_1^4$

A normal field on a spacelike surface in $R^4_1$ is called bi-normal if $K^ν$, the determinant of Weingarten map associated with $ν$, is zero. In this paper we give a relationship between the spacelike pseudo-planar surfaces and spacelike pseudo-umbilical surfaces, then study the bi-normal fields on spacelike ruled surfaces and spacelike surfaces of revolution.