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 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…
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 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…
In this article we fully classify regular tubular surfaces in Euclidean, Lorentzian and hyperbolic 3-spaces whose Gaussian and mean curvatures $K$ and $H$ verify a polynomial relation. More precisely, we determine the set $S(Q)$ of all regular tubular surfaces whose curvatures verify a given polynomial relation $Q(K,H)=0$, and the set $Q(S)$ of all polynomial relations vanished by the curvatures of a given tubular surface $S$.
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…
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…
We prove the existence of a complete locally Lipschitz continuous hypersurface in weak sense with prescribed Weingarten curvature and asymptotic boundary at infinity in hyperbolic space under certain assumptions.
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 …
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…
In this paper we study the linear Weingarten equation defined by the fully non-linear PDE $$a\, \mbox{div}\frac{Du}{\sqrt{1+|Du|^2}}+b\, \frac{\mbox{det}D^2u}{(1+|Du|^2)^2}=ϕ\left(\frac{1}{\sqrt{1+|Du|^2}}\right)$$ in a domain $Ω\subset\mathbb{R}^2$, where $ϕ\in C^1([-1,1])$ and $a,b\in\mathbb{R}$. We approach the existence of radial solutions when $Ω$ is a disk of small radius, giving an affirmative answer when the PDE is of elliptic type. In the hyperbolic case we show that no radial solution exists, while in the parabolic case we find explicitly all the solutions. Finally, in the elliptic c…
We investigate on the existence of smooth complete hypersurface with prescribed Weingarten curvature and asymptotic boundary at infinity in hyperbolic space under the assumption that there exists an asymptotic subsolution. We give an affirmative answer for the case $k = n$ when the asymptotic boundary $Γ$ bounds a uniformly convex domain, and for $k < n$ when $Γ$ bounds a disk, utilizing Pogorelov type interior second order estimate. Our result complements our previous work \cite{Sui2019, Sui-Sun}, and generalizes the asymptotic Plateau type problem to non-constant prescribed curvature case.
In this paper, we can obtain curvature estimates for spacelike admissible graphic hypersurfaces in the $(n+1)$-dimensional Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$, and through which the existence of spacelike admissible graphic hypersurfaces, with prescribed $2$-th Weingarten curvature and Dirichlet boundary data, defined over a strictly convex domain in the hyperbolic plane $\mathscr{H}^{n}(1)\subset\mathbb{R}^{n+1}_{1}$ of center at origin and radius $1$, can be proven.
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…
In the present paper, firstly we obtain the general expression of the canal hypersurfaces which are formed as the envelope of a family of pseudo hyperspheres, pseudo hyperbolic hyperspheres and null hypercones in $E_{1}^{4}$ and give their some geometric invariants such as unit normal vector fields, Gaussian curvatures, mean curvatures and principal curvatures. Also we give some results about their flatness and minimality conditions and Weingarten canal hypersurfaces. Also, we obtain these characterizations for tubular hypersurfaces in $E_{1}^{4}$ by taking constant radius function and finally…
