Filippo Morabito, M. Magdalena Rodriguez
In this paper we finish the classification of rotational special Weingarten surfaces in S^2 x R and H^2 x R; i.e. rotational surfaces in S^2 x R and H^2 x R whose mean curvature h and extrinsic curvature K_e satisfy h=f(h^2-K_e), for some function f in C^1([0,+infty)) such that 4x(f'(x))^2<1 for any x>=0.
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We prove that any uniformly elliptic Weingarten (topological) sphere in S2xR must be congruent to the canonical example associated to the Weingarten equation. The result is obtained by proving that rotational uniformly elliptic Weingarten surfaces in S2xR have bounded second fundamental form together with a Hopf type result by J. A. Gálvez and P. Mira.
In this paper, we study the rotational surfaces in the isotropic 3-space I^3. satisfying Weingarten conditions in terms of the relative curvature K (analogue of the Gaussian curvature) and the isotropic mean curvature H. In particular, we classify such surfaces of linear Weingarten type in I^3.
The aim of this paper is to present a complete description of all rotational linear Weingarten surface into the Euclidean sphere S3. These surfaces are characterized by a linear relation aH+bK=c, where H and K stand for their mean and Gaussian curvatures, respectively, whereas a; b and c are real constants.
In this article, we study spacelike and timelike rotational surfaces in a 3--dimensional de Sitter space $\mathbb{S}^3_1$ which are the orbit of a regular curve under the action of the orthogonal transformation of 4--dimensional Minkowski space $\mathbb{E}^4_1$ leaving a spacelike, a timelike or a degenerate plane pointwise fixed. We determine the profile curve of such Weingarten rotational surfaces parameterized by the principal curvature. Then, we classify spacelike and timelike Weingarten rotational surface in $\mathbb{S}^3_1$ with the principal curvatures $κ$ and $λ$ satisfying $κ=aλ+b$ or…
Rotational Weingarten surfaces in Lorentz-Minkowski space
本文提出了一种基于几何线动量的研究方法，用于分类洛伦兹-闵可夫斯基空间中的旋转魏因加滕表面，并引入了非退化旋转二次曲面的表征。
Weingarten surfaces are those whose principal curvatures satisfy a functional relation, whose set of solutions is called the curvature diagram or the W-diagram of the surface. Making use of the notion of geometric linear momentum of a plane curve, we propose a new approach to the study of rotational Weingarten surfaces in Euclidean 3-space. Our contribution consists of reducing any type of Weingarten condition on a rotational surface to a first order differential equation on the momentum of the generatrix curve. In this line, we provide two new classification results involving a cubic and an h…
In this paper we review some author's results about Weingarten surfaces in Euclidean space $\r^3$ and hyperbolic space $\h^3$. We stress here in the search of examples of linear Weingarten surfaces that satisfy a certain geometric property. First, we consider Weingarten surfaces in $\r^3$ that are foliated by circles, proving that the surface is rotational, a Riemann example or a generalized cone. Next we classify rotational surfaces in $\r^3$ of hyperbolic type showing that there exist surfaces that are complete. Finally, we study linear Weingarten surfaces in $\h^3$ that are invariant by a g…
A linear Weingarten surface in Euclidean space ${\bf R}^3$ is a surface whose mean curvature $H$ and Gaussian curvature $K$ satisfy a relation of the form $aH+bK=c$, where $a,b,c\in {\bf R}$. Such a surface is said to be hyperbolic when $a^2+4bc<0$. In this paper we classify all rotational linear Weingarten surfaces of hyperbolic type. As a consequence, we obtain a family of complete hyperbolic linear Weingarten surfaces in ${\bf R}^3$ that consists into periodic surfaces with self-intersections.
We demonstrate that every non-tubular channel linear Weingarten surface in Euclidean space is a surface of revolution, hence parallel to a catenoid or a rotational surface of non-zero constant Gauss curvature. We provide explicit parametrizations and deduce existence of complete hyperbolic linear Weingarten surfaces.
In this paper, we show that an embedded Weingarten surface in S^3 of genus 1 must be rotationally symmetric, provided that certain structure conditions are satisfied. The argument involves an adaptation of our proof of Lawson's Conjecture for minimal tori.
Height estimate for special weingarten surfaces of elliptic type in warped product
研究扭曲乘积空间中紧致椭圆型特殊 Weingarten 曲面的几何问题，核心贡献是建立垂直高度估计。
We show by phase space analysis that there are exactly 17 possible qualitative behaviors for a rotational surface in $\mathbb{R}^3$ that satisfies an arbitrary elliptic Weingarten equation $W(κ_1,κ_2)=0$, and study the singularities of such examples. As global applications of this classification, we prove a sharp halfspace theorem for general elliptic Weingarten equations of finite order, and a classification of peaked elliptic Weingarten spheres with at most two singularities. In the case that $W$ is not elliptic, we give a negative answer to a question by Yau regarding the uniqueness of rota…
Let $M^n$ be either a simply connected space form or a rank-one symmetric space of noncompact type. We consider Weingarten hypersurfaces of $M\times\mathbb R$, which are those whose principal curvatures $k_1,\dots ,k_n$ and angle function $θ$ satisfy a relation $W(k_1,\dots,k_n,θ^2)=0,$ being $W$ a differentiable function which is symmetric with respect to $k_1,\dots, k_n.$ When $\partial W/\partial k_i>0$ on the positive cone of $\mathbb R^n,$ a strictly convex Weingarten hypersurface determined by $W$ is said to be elliptic. We show that, for a certain class of Weingarten functions $W,$ ther…
Lie minimal Weingarten surfaces
本文研究黎曼空间形式中 Lie 球面不变能量的临界点 Lie 极小曲面，通过 Euler-Lagrange 方程刻画其主曲率，并证明常平均曲率曲面及平坦外围空间中满足仿射关系的曲面和椭圆线性 Weingarten 曲面均具有旋转性。
Special Weingarten surfaces with planar convex boundary
本研究在特殊Weingarten曲面框架下证明了Ros-Rosenberg定理，得出在温和假设下三维欧氏空间中带平面凸边界的紧致连通嵌入特殊Weingarten曲面是拓扑圆盘的结论。
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…
In this paper we study surfaces in Euclidean 3-space foliated by pieces of circles and that satisfy a Weingarten condition of type $a H+b K=c$, where $a,b$ and $c$ are constant and $H$ and $K$ denote the mean curvature and the Gauss curvature respectively. We prove that a such surface must be a surface of revolution, a Riemann minimal surface or a generalized cone.
Rotational cmc surfaces in terms of Jacobi elliptic functions
本文利用雅可比椭圆函数和Lie球面几何方法，对非欧空间形式中的旋转常平均曲率曲面进行分类，并给出双曲空间中旋转常调和平均曲率曲面的显式参数化，补全了管道线性Weingarten曲面分类的最后空白。
In this article, we study complete surfaces $Σ$, isometrically immersed in the product space $\mathbb{H}^2\times\mathbb{R}$ or $\mathbb{S}^2\times\mathbb{R}$ having positive extrinsic curvature $K_e$. Let $K_i$ denote the intrinsic curvature of $Σ$. Assume that the equation $aK_i+bK_e=c$ holds for some real constants $a\neq0$, $b>0$ and $c$. The main result of this article state that when such a surface is a topological sphere it is rotational.
In this paper, under suitable settings, we can obtain the existence of solutions to a class of prescribed Weingarten curvature equations in warped product manifolds of special type by the standard degree theory based on the a priori estimates for the solutions. This is to say that the existence of closed hypersurface (which is graphic with respect to the base manifold and whose $k$-th Weingarten curvature satisfies some constraint) in a given warped product manifold of special type can be assured.
Given $a,b\in\mathbb{R}$ and $Φ\in C^1(\mathbb{S}^2)$, we study immersed oriented surfaces $Σ$ in the Euclidean 3-space $\mathbb{R}^3$ whose mean curvature $H$ and Gauss curvature $K$ satisfy $2aH+bK=Φ(N)$, where $N:Σ\rightarrow\mathbb{S}^2$ is the Gauss map. This theory widely generalize some of paramount importance such as the ones constant mean and Gauss curvature surfaces, linear Weingarten surfaces and self-translating solitons of the mean curvature flow. Under mild assumptions on the prescribed function $Φ$, we exhibit a classification result for rotational surfaces in the case that the …
Properties and Transformations of Weingarten Surfaces
本文研究E3中旋转对称Weingarten曲面，分析脐点斜率限制、SL2(R)几何变换及Lagrangian变分表述，并证明半二次型曲面上作用的传递性与稳定性。
Discrete linear Weingarten surfaces in space forms are characterized as special discrete $Ω$-nets, a discrete analogue of Demoulin's $Ω$-surfaces. It is shown that the Lie-geometric deformation of $Ω$-nets descends to a Lawson transformation for discrete linear Weingarten surfaces, which coincides with the well-known Lawson correspondence in the constant mean curvature case.
We prove that any strongly regular Weingarten surface in Euclidean space carries locally geometric principal parameters. The basic theorem states that any strongly regular Weingarten surface is determined up to a motion by its structural functions and the normal curvature function satisfying a geometric differential equation. We apply these results to the special Weingarten surfaces: minimal surfaces, surfaces of constant mean curvature and surfaces of constant Gauss curvature.
In this article, we shall derive by elementary calculations the Gauss map, spherical image, Weingarten map and the curvatures at identity of the special linear group, that is, the matrices of determinant 1. We could not find any reference where this simple, nice computation has been explicitly put down.
Elliptic special Weingarten surfaces of minimal type in $\mathbb{R}^3$ of finite total curvature
研究\(\mathbb{R}^3\)中有限全曲率的完备连通嵌入极小类型椭圆特殊Weingarten曲面，利用渐近理论推导公式，用Alexandrov反射方法证明定理，确定其旋转对称性，回答相关问题，指出特定条件下此类曲面仅为平面和特殊悬链面。
In the work \cite{Laredo} the author shows that every hypersurface in Euclidean space is locally associated to the unit sphere by a sphere congruence, whose radius function $R$ is a geometric invariant of hypersurface. In this paper we define for any surface $Σ$ its spherical mean curvature $H_S$ which depends on principal curvatures of $Σ$ and the radius function $R$. Then we consider two classes of surfaces: the ones with $H_S = 0$, called $H_1$-surfaces, and the surfaces with spherical mean curvature of harmonic type, named $H_2$-surfaces. We provide for each these classes a Weierstrass-typ…
Some structure theorems for Weingarten surfaces
研究边界为凸平面曲线的Weingarten曲面，给出曲面继承边界对称性的充分条件，并推广常平均曲率曲面的相关结果。
Elliptic Weingarten surfaces of minimal type in $\mathbb{R} \times_{h} \mathbb{R}$
本文在径向对称扭曲乘积 $\mathbb{R} \times_h \mathbb{R}$ 中研究满足 $H=f(H^2-K)$ 的极小类型椭圆 Weingarten 曲面，证明旋转不变例子的存在唯一性并分析生成曲线。
In this work we study surfaces in radial conformally flat spaces. We characterize surfaces of rotation with constant Gaussian and Extrinsic curvature in these radial 3-spaces. We prove that all the spheres in the conformal 3-space have constant Gaussian curvature $K=1$ if, and only if, the conformal factor is special. In this special case we study geometric properties of this ambient 3-space, and as an application we prove that it is isometric to the space ${\mathbb{S}}^2\times {\mathbb{R}}$, so we consider it as the {\em Radial Model} of ${\mathbb{S}}^2\times {\mathbb{R}}$. We obtain two clas…
We study time-like surfaces in the three-dimensional Minkowski space with diagonalizable second fundamental form. On any time-like W-surface we introduce locally natural principal parameters and prove that such a surface is determined uniquely (up to motion) by a special invariant function, which satisfies a natural non-linear partial differential equation. This result can be interpreted as a solution of the Lund-Regge reduction problem for time-like W-surfaces with real principal curvatures in Minkowski space. We apply this theory to linear fractional time-like W-surfaces and obtain the natur…
On any space-like W-surface in the three-dimensional Minkowski space we introduce locally natural principal parameters and prove that such a surface is determined uniquely up to motion by a special invariant function, which satisfies a natural non-linear partial differential equation. This result can be interpreted as a solution to the Lund-Regge reduction problem for space-like W-surfaces in Minkowski space. We apply this theory to linear fractional space-like W-surfaces and obtain the natural non-linear partial differential equations describing them. We obtain a characterization of space-lik…
The Renormalized Area--Willmore Relation
该研究将Alexakis与Mazzeo关于双曲空间中极小曲面的重整化面积与Willmore能量的关系结果推广到更广的几何方程类，证实存在满足普适比例关系的其他几何曲面方程，并对旋转线性Weingarten曲面给出相关积分表示与Willmore能量性质。
In the 1950's Hopf gave examples of non-round convex 2-spheres in Euclidean 3-space with rotational symmetry that satisfy a linear relationship between their principal curvatures. In this paper we investigate conditions under which evolving a smooth rotationally symmetric sphere by a linear combination of its radii of curvature yields a Hopf sphere. When the coefficients of the flow have certain integer values, the fate of an initial sphere is entirely determined by the local geometry of its isolated umbilic points. A surprising variety of behaviours is uncovered: convergence to round spheres …
In this work, we consider spacelike surfaces in Minkowski space $\hbox{\bf E}%_{1}^{3}$ that satisfy a linear Weingarten condition of type $κ_{1}=mκ_{2}+n$, where $m$ and $n$ are constant and $κ_{1}$ and $κ_{2}$ denote the principal curvatures at each point of the surface. We study the family of surfaces foliated by a uniparametric family of circles in parallel planes. We prove that the surface must be rotational or the surface is part of the family of Riemann examples of maximal surfaces ($m=-1$, $n=0$). Finally, we consider the class of rotational surfaces for the case $n=0$, obtaining a fir…
For a two-dimensional surface in the four-dimensional Euclidean space we introduce an invariant linear map of Weingarten type in the tangent space of the surface, which generates two invariants k and kappa. The condition k = kappa = 0 characterizes the surfaces consisting of flat points. The minimal surfaces are characterized by the equality kappa^2=k. The class of the surfaces with flat normal connection is characterized by the condition kappa = 0. For the surfaces of general type we obtain a geometrically determined orthonormal frame field at each point and derive Frenet-type derivative form…
We study surface energies depending on the mean curvature in total spaces of Killing submersions, which extend the classical notion of Willmore energy. Based on a symmetry reduction procedure, we construct vertical tori critical for these mean curvature energies. These vertical tori are based on closed curves critical for curvature energy functionals in Riemannian 2-space forms. The binormal evolution of these critical curves in Riemannian 3-space forms generates rotational tori solutions for an ample family of Weingarten surfaces. Therefore, we also introduce some correspondence results betwe…
On intrinsic rotational surfaces in the Lorentz-Minkowski space
研究Lorentz-Minkowski空间中的内蕴旋转曲面，通过替换平均曲率常值性为Weingarten自同态的特定表达式，证明平均曲率为常数且旋转角是线性函数，结果涵盖类空/类时情况，还确定了零平均曲率下的相关曲面族。
We consider spacelike surfaces in the four-dimensional Minkowski space and introduce geometrically an invariant linear map of Weingarten-type in the tangent plane at any point of the surface under consideration. This allows us to introduce principal lines and an invariant moving frame field. Writing derivative formulas of Frenet-type for this frame field, we obtain eight invariant functions. We prove a fundamental theorem of Bonnet-type, stating that these eight invariants under some natural conditions determine the surface up to a motion. We show that the basic geometric classes of spacelike …
On the Existence of Closed Biconservative Surfaces in Space Forms
本文将非 CMC 双保守旋转曲面的轮廓曲线刻画为曲率能量临界曲线，并据此证明三维球面中存在离散双参数族的闭合非 CMC 双保守曲面，但它们均非嵌入。
Characterizations of Loxodromes on Rotational Surfaces in Euclidean 3--Space
本文研究了欧几里得3空间中旋转曲面上洛克斯多形的几何特性，探讨了其在不同曲面下的参数化、曲率与挠率计算，并揭示了洛克斯多形在特定条件下的渐近曲线特性。
General rotational surfaces as a source of examples of surfaces in the four-dimensional Euclidean space have been introduced by C. Moore. In this paper we consider the analogue of these surfaces in the Minkowski 4-space. On the base of our invariant theory of spacelike surfaces we study general rotational surfaces with special invariants. We describe analytically the flat general rotational surfaces and the general rotational surfaces with flat normal connection. We classify completely the minimal general rotational surfaces and the general rotational surfaces consisting of parabolic points.
In the present paper we study the global behaviour of geodesics on a Randers metric, defined on a topological cylinder, obtained as the solution of the Zermelo's navigation problem. Our wind is not necessarily a Killing field. In special we concentrate our study on the geodesics equation on the base manifold, the conjugate and cut loci.
We develop an invariant local theory of Lorentz surfaces in pseudo-Euclidean 4-space by use of a linear map of Weingarten type. We find a geometrically determined moving frame field at each point of the surface and obtain a system of geometric functions. We prove a fundamental existence and uniqueness theorem in terms of these functions. On any Lorentz surface with parallel normalized mean curvature vector field we introduce special geometric (canonical) parameters and prove that any such surface is determined up to a rigid motion by three invariant functions satisfying three natural partial d…
Submanifolds of coordinate finite-type were introduced in HV1. A submanifold of a Euclidean space is called a coordinate finite-type submanifold if its coordinate functions are eigenfunctions of Δ. In the present study we consider coordinate finite-type surfaces in E^4. We give necessary and sufficient conditions for generalized rotation surfaces in E^4 to become coordinate finite-type. We also give some special examples.
We classify constant mean curvature surfaces invariant by a 1-parameter group of isometries in the Berger spheres and in the special linear group Sl(2, R). In particular, all constant mean curvature spheres in those spaces are described explicitly, proving that they are not always embedded. Besides new examples of Delaunay-type surfaces are obtained. Finally the relation between the area and volume of these spheres in the Berger spheres is studied, showing that, in some cases, they are not solution to the isoperimetric problem.
We construct a special class of Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric which are one-parameter systems of meridians of rotational hypersurfaces with lightlike axis and call them meridian surfaces. We give the complete classification of the meridian surfaces with constant Gauss curvature and prove that there are no meridian surfaces with parallel mean curvature vector field other than CMC surfaces lying in a hyperplane. We also classify the meridian surfaces with parallel normalized mean curvature vector field. We show that in the family of the meridian surfaces th…
Meridian surfaces in the Euclidean 4-space are two-dimensional surfaces which are one-parameter systems of meridians of a standard rotational hypersurface. On the base of our invariant theory of surfaces we study meridian surfaces with special invariants. In the present paper we give the complete classification of Chen meridian surfaces and meridian surfaces with parallel normal bundle.
Loxodromes and geodesics on rotational surfaces in pseudo-isotropic space
本文首次研究了伪迷向空间 I_p^3 中的恒向线，定义了伪迷向角，并推导出旋转曲面上类空与类时恒向线及测地线的方程。
We obtain a classification result for rotational surfaces in the Heisenberg space and the universal cover of the special linear group, whose mean curvature is given as a prescribed $C^1$ function depending on their angle function. We show that these surfaces behave like the Delaunay surfaces of constant mean curvature, under some assumptions on the prescribed function. In contrast with the constant mean curvature case, we exhibit the existence of rotational, embedded tori, providing counterexamples of the Alexandrov problem for this class of immersed surfaces.
In the present article we study a special class of surfaces in the four-dimensional Euclidean space, which are one-parameter systems of meridians of the standard rotational hypersurface. They are called meridian surfaces. We classified semi-parallel meridian surface in 4-dimensional Euclidean space $E^4$.
A Liouville-Weierstrass correspondence for Spacelike and Timelike Minimal Surfaces in $\mathbb{L}^3$
本文研究了Liouville方程与空间状和时间状最小曲面在$\mathbb{L}^3$中的对应关系，通过复分析和para复分析提供了统一处理方法，并给出了显式例子。
We consider a special class of spacelike surfaces in the Minkowski 4-space which are one-parameter systems of meridians of the rotational hypersurface with timelike or spacelike axis. We call these surfaces meridian surfaces of elliptic or hyperbolic type, respectively. On the base of our invariant theory of surfaces we study meridian surfaces with special invariants and give the complete classification of the meridian surfaces with constant Gauss curvature or constant mean curvature. We also classify the Chen meridian surfaces and the meridian surfaces with parallel normal bundle.
We consider an infinite 3-dimensional elastic continuum whose material points experience no displacements, only rotations. This framework is a special case of the Cosserat theory of elasticity. Rotations of material points are described mathematically by attaching to each geometric point an orthonormal basis which gives a field of orthonormal bases called the coframe. As the dynamical variables (unknowns) of our theory we choose the coframe and a density. We write down the general dynamic variational functional for our rotational theory of elasticity, assuming our material to be physically lin…
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