arXiv research

Uniformly elliptic Weingarten spheres in S2xR are congruent to a canonical example.

Study of surfaces in space forms using Lie sphere geometry.

The paper proves spheres in certain 3-manifolds are always rotational.

This study classifies quadric surfaces in 3-sphere as Weingarten surfaces.

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.

The study examines hypersurfaces in warped products and their properties.

Study of 17 surface behaviors and singularities for elliptic Weingarten equations.

The study explores special hypersurfaces in Riemannian products, focusing on elliptic Weingarten conditions.

Researchers classify cmc surfaces using Jacobi elliptic functions.

Lie minimal surfaces are characterized by differential equations of principal curvatures.

We present some results on the boundedness of the mean curvature of proper biharmonic submanifolds in spheres. A partial classification result for proper biharmonic submanifolds with parallel mean curvature vector field in spheres is obtained. Then, we completely classify the proper biharmonic submanifolds in spheres w…

We show that any star-shaped convex hypersurface with constant Weingarten curvature in the deSitter-Schwarzschild manifold is a sphere of symmetry. Moreover, we study an isoperimetric problem for bounded domains in the doubled Schwarzschild manifold. We prove the existence of an isoperimetric surface for any value of t…

In this paper we develop an abstract theory for the Codazzi equation on surfaces, and use it as an analytic tool to derive new global results for surfaces in the space forms ${\bb R}^3$, ${\bb S}^3$ and ${\bb H}^3$. We give essentially sharp generalizations of some classical theorems of surface theory that mainly depen…

We prove a uniqueness theorem for immersed spheres of prescribed (non-constant) mean curvature in homogeneous three-manifolds. In particular, this uniqueness theorem proves a conjecture by A.D. Alexandrov about immersed spheres of prescribed Weingarten curvature in R3 for the special but important case of prescribed me…

We classify the self-similar solutions to a class of Weingarten curvature flow of connected compact convex hypersurfaces, isometrically immersed into space forms with non-positive curvature, and obtain a new characterization of a sphere in a Euclidean space $\R^{n+1}$.

Compact Special Weingarten surfaces with planar convex boundaries are disks.

In this study, we investigated the (K,H), (K,K_{II}), (H,K_{II})-Weingarten and (K,H),(K,K_{II}),(H,K_{II}) and (K,H,K_{II})-linear Weingarten canal surfaces in IR^3.

New method classifies HCMU surfaces in 3D space forms as Weingarten surfaces.

Study asymptotic behavior of Weingarten surfaces at infinity.

New approach classifies rotational Weingarten surfaces in Lorentz-Minkowski space.

Paper proves existence of solutions to curvature equations.

We prove that the natural principal parameters on a given Weingarten surface are also natural principal parameters for the parallel surfaces of the given one. As a consequence of this result we obtain that the natural PDE of any Weingarten surface is the natural PDE of its parallel surfaces. We show that the linear fra…

In this paper we use the relationship between conformal metrics on the sphere and horospherically convex hypersurfaces in the hyperbolic space for giving sufficient conditions on a conformal metric to be radial under some constrain on the eigenvalues of its Schouten tensor. Also, we study conformal metrics on the spher…

Solves an old problem by showing round spheres are the only compact surfaces with specific curvature properties.

Efficiently estimates Weingarten maps and curvatures from manifold data.

Authors compute Weingarten map and curvatures for SL(n, R).

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 eq…

Study on properties and transformations of Weingarten surfaces in 3D space.

The study proves planes are the only complete uniformly elliptic Weingarten multigraphs.

In this paper, we study meridian surfaces of Weingarten type in Euclidean 4-space E^4. We give the neccessary and sufficient conditions for a meridian surface in E^4 to become Weingarten type.

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, …

An idea of Hopf's for applying complex analysis to the study of constant mean curvature spheres is generalized to cover a wider class of spheres, namely, those satisfying a Weingarten relation of a certain type, namely H = f(H^2-K) for some smooth function f, where H and K are the mean and Gauss curvatures, respectivel…

Estimates heights of special surfaces in warped products.

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.

Paper solves Christoffel-Minkowski and Weingarten curvature problems in hyperbolic space.

In this paper we will classify those translation surfaces in E3 involving polynomials which are Weingarten surfaces. We analyze Weingarten translation surfaces satisfying 2aH + bK = 0. We study also other types of translation surfaces, involving power functions, for which the second Gaussian curvature vanishes.

The paper describes and classifies surfaces in isotropic 3-space.

Study classifies semi-discrete linear Weingarten surfaces with Weierstrass-type representations and analyzes their singularities.

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.

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…

We study flows of hypersurfaces in Riemannian manifolds with specific curvature speeds.

Extends theory of minimal surfaces to elliptic special Weingarten surfaces.

In this paper, we study the timelike tubular Weingarten surfaces in 3-dimensional Minkowski space $IR_1^3$.We have obtained some conditions for being $({K_{II},H})$, $({K_{II},K})$, timelike tubular Weingarten surfaces where are the second Gaussian curvature the Gaussian curvature and the mean curvature, respectively.

The paper explores conditions for symmetries in Weingarten surfaces.

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 L…

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. …

Study on singularities of discrete Weingarten surfaces in Riemannian and Lorentzian spaceforms.

New closed non-CMC biconservative surfaces found in round 3-sphere.