A surface in hyperbolic space $\h^3$ invariant by a group of parabolic isometries is called a parabolic surface. In this paper we investigate parabolic surfaces of $\h^3$ that satisfy a linear Weingarten relation of the form $aκ_1+bκ_2=c$ or $aH+bK=c$, where $a,b,c\in \r$ and, as usual, $κ_i$ are the principal curvatures, $H$ is the mean curvature and $K$ is de Gaussian curvature. We classify all parabolic linear Weingarten surfaces in hyperbolic space.
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…
Weingarten transformations which, by definition, preserve the asymptotic lines on smooth surfaces have been studied extensively in classical differential geometry and also play an important role in connection with the modern geometric theory of integrable systems. Their natural discrete analogues have been investigated in great detail in the area of (integrable) discrete differential geometry and can be traced back at least to the early 1950s. Here, we propose a canonical analogue of (discrete) Weingarten transformations for hyperbolic nets, that is, C^1-surfaces which constitute hybrids of sm…
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 $S$.
We define discrete flat surfaces in hyperbolic 3-space from the perspective of discrete integrable systems and prove properties that justify the definition. We show how these surfaces correspond to previously defined discrete constant mean curvature 1 surfaces in hyperbolic 3-space, and we also describe discrete focal surfaces (discrete caustics) that can be used to define singularities on discrete flat surfaces. We also examine discrete linear Weingarten surfaces of Bryant type in hyperbolic 3-space, and consider an example of a discrete flat surface related to the Airy equation that exhibits…
We study 2-dimensional submanifolds of the space ${\mathbb{L}}({\mathbb{H}}^3)$ of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. Such a surface is Lagrangian iff there exists a surface in ${\mathbb{H}}^3$ orthogonal to the geodesics of $Σ$. We prove that the induced metric on a Lagrangian surface in ${\mathbb{L}}({\mathbb{H}}^3)$ has zero Gauss curvature iff the orthogonal surfaces in ${\mathbb{H}}^3$ are Weingarten: the eigenvalues of the second fundamental form are functionally related. We then classify the totally null surfaces in ${\mathbb{L…
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 prove several topological properties of linear Weingarten surfaces of Bryant type, as wave fronts in hyperbolic 3-space. For example, we show the orientability of such surfaces, and also co-orientability when they are not flat. Moreover, we show an explicit formula of the non-holomorphic hyperbolic Gauss map via another hyperbolic Gauss map which is holomorphic. Using this, we show the orientability and co-orientability of CMC-1 faces (i.e., constant mean curvature one surfaces with admissible singular points) in de Sitter 3-space. (CMC-1 faces might not be wave fronts in general, but belon…
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.
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 …
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 study parabolic linear Weingarten surfaces in hyperbolic space $\rlopezh^3$. In particular, we classify two family of parabolic surfaces: surfaces with constant Gaussian curvature and surfaces that satisfy the relation $aκ_1+bκ_2=c$, where $κ_i$ are the principal curvatures, and $a,b$ and $c$ are constant.
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 surfaces with constant harmonic-mean curvature one (HMC-1 surfaces) in hyperbolic three-space. We allow them to have certain kinds of singularities, and discuss some global properties. As well as flat surfaces and surfaces with constant mean curvature one (CMC-1 surfaces), HMC-1 surfaces belong to a certain class of Weingarten surfaces. From the viewpoint of parallel surfaces, CMC-1 surfaces and HMC-1 surfaces are representative among this class.
In this paper we develop a global correspondence between immersed horospherically convex hypersurfaces in hyperbolic space and complete conformal metrics on domains in the sphere. We establish results on when the hyperbolic Gauss map is injective and when an immersed horospherically convex hypersurface can be unfolded along the normal flow into an embedded one. These results allow us to establish general Alexandrov reflection principles of elliptic problems of both Weingarten hypersurfaces and complete conformal metrics and relations between them. Consequently, we are able to obtain, for insta…
We derive a relationship between the eigenvalues of the Weyl-Schouten tensor of a conformal representative of the conformal infinity of a hyperbolic Poincaré manifold and the principal curvatures on the level sets of its uniquely associated defining function with calculations based on [9] [10]. This relationship generalizes the result for hypersurfaces in ${\H}^{n+1}$ and their connection to the conformal geometry of ${\SS}^n$ as exhibited in [7] and gives a correspondence between Weingarten hypersurfaces in hyperbolic Poincaré manifolds and conformally invariant equations on the conformal inf…
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.
Let $(M,g)$ be an asymptotically hyperbolic manifold with a smooth conformal compactification. We establish a general correspondence between semilinear elliptic equations of scalar curvature type on $\del M$ and Weingarten foliations in some neighbourhood of infinity in $M$. We focus mostly on foliations where each leaf has constant mean curvature, though our results apply equally well to foliations where the leaves have constant $σ_k$-curvature. In particular, we prove the existence of a unique foliation near infinity in any quasi-Fuchsian 3-manifold by surfaces with constant Gauss curvature.…
In this note, we establish the dihedral rigidity phenomenon for a collection of parabolic polyhedrons enclosed by horospheres in hyperbolic manifolds, extending Gromov's comparison theory to metrics with negative scalar curvature lower bounds. Our result is a localization of the positive mass theorem for asymptotically hyperbolic manifolds. We also motivate and formulate some open questions concerning related rigidity phenomenon and convergence of metrics with scalar curvature lower bounds.
We consider classical curvature flows: 1-parameter families of convex embeddings of the 2-sphere into Euclidean 3-space which evolve by an arbitrary (non-homogeneous) function of the radii of curvature. The associated flow of the radii of curvature is a second order system of partial differential equations which we show decouples to highest order. We determine the conditions for this system to be parabolic and investigate the lower order terms. The zeroth order terms are shown to form a Hamiltonian system, which is therefore completely integrable. We find conditions on parabolic curvature flow…
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 …
We investigate the geometric properties of marginally trapped surfaces (surfaces which have null mean curvature vector) in the spaces of oriented geodesics of Euclidean 3-space and hyperbolic 3-space, endowed with their canonical neutral Kaehler structures. We prove that every rank one surface in these four manifolds is marginally trapped. In the Euclidean case we show that Lagrangian rotationally symmetric sections are marginally trapped and construct an explicit family of marginally trapped Lagrangian tori. In the hyperbolic case we explore the relationship between marginally trapped and Wei…
