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 fractional Weingarten surfaces are exactly the surfaces satisfying a linear relation between their three curvatures. Our main result is classification of the natural PDE's of Weingarten surfaces with linear relation between their curvatures.
In this paper, we consider a class of prescribed Weingarten curvature equations. Under some sufficient condition, we obtain an existence result by the standard degree theory based on the a prior estimates for the solutions to the prescribed Weingarten curvature equations.
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.
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.
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 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…
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.
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 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 \cite{WeiWu22-3} (J.Geom.Anal.32, 199(2022)), we classified HCMU surfaces in 3-dimensional Euclidean space as Weingarten surfaces by holomorphic functions. In this paper, using a totally different method from \cite{WeiWu22-3}, we will classify HCMU surfaces in 3-dimensional space forms as Weingarten surfaces. Moreover we will give a criteria of a Weingarten surface which is also an HCMU surface.
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.
We give a relatively simple proof that a translation surface in Euclidean space that satisfies a relation of type $aH+bK=c$, for some real numbers $a,b,c$, where $H$ and $K$ are the mean curvature and the Gauss curvature of the surface, respectively, must have $a=0$ or $b=0$, and thus, $K$ is constant or $H$ is constant. Our method of proof extends to the Lorentzian ambient space.
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 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.
In this study, we consider canal surfaces according to parallel transport frame in Euclidean space $\mathbb{E}^{4}$. The curvature properties of these surfaces are investigated with respect to $k_{1}$, $k_{2}$ and $k_{3}$ which are principal curvature functions according to parallel transport frame. We also give an example of canal surfaces in $\mathbb{E}^{4}.$ Further, we point out that if spine curve $γ$ is a straight line, then $M$ is a Weingarten canal surface and also $M$ is a linear Weingarten tube surface. Finally, the visualization of the projections of canal surfaces in $\mathbb{E}^{3…
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 show that compact embedded starshaped $r$-convex hypersurfaces of certain warped products satisfying $H_r=aH+b$ with $a\geqslant 0$, $b>0$, where $H$ and $H_r$ are respectively the mean curvature and $r$-th mean curvature is a slice. In the case of space forms, we show that without the assumption of starshapedness, such Weingarten hypersurfaces are geodesic spheres. Finally, we prove that, in the case of space forms, if $Hr-aH-b$ is close to $0$ then the hypersurface is close to geodesic sphere for the Hausdorff distance. We also prove an anisotropic version of this stability result in the …
In this paper, we prove the existence of hypersurfaces in the Euclidean space with prescribed boundary and whose k-th Weingarten curvature equals a given function that depends on the normal of the hypersurface. The proof is based on the solvability of a fully nonlinear elliptic PDE. The required a priori estimates are established under the natural assumptions that the prescribed boundary is strictly convex and the prescribed function satisfies a Serrin type condition.
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…
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 introduce a new class of surfaces in Euclidean $3$-space, called surfaces of osculating circles, using the concept of osculating circle of a regular curve. These surfaces contain a uniparametric family of planar lines of curvature. In this paper, we classify those ones that are canal surfaces and Weingarten surfaces.
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…
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…
We prove that any complete, uniformly elliptic Weingarten surface in Euclidean $3$-space whose Gauss map image omits an open hemisphere is a cylinder or a plane. This generalizes a classical theorem by Hoffman, Osserman and Schoen for constant mean curvature surfaces. In particular, this proves that planes are the only complete, uniformly elliptic Weingarten multigraphs. We also show that this result holds for a large class of non-uniformly elliptic Weingarten equations. In particular, this solves in the affirmative the Bernstein problem for entire graphs for that class of elliptic equations. …
In the present paper, firstly we obtain the general expression of canal hypersurfaces in Euclidean n-space and deal with canal hypersurfaces in Euclidean 4-space E4. We compute Gauss map, Gaussian curvature and mean curvature of canal hypersurfaces in E4 and obtain an important relation between the mean and Gaussian curvatures as 3Hrho = Krho^3-2. We prove that, the flat canal hypersurfaces in Euclidean 4-space are only circular hypercylinders or circular hypercones and minimal canal hypersurfaces are only generalized catenoids. Also, we state the expression of tubular hypersurfaces in Euclide…
We investigate some characteristic properties of specific Weingarten surfaces in the three-dimensional Euclidean space using the nets of the lines of curvature resp. the asymptotic lines on both central surfaces of them.
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 the present paper, a new type of ruled surfaces called osculating-type (OT)-ruled surface is introduced and studied. First, a new orthonormal frame is defined for OT-ruled surfaces. The Gaussian and the mean curvatures of these surfaces are obtained and the conditions for an OT-surface to be flat or minimal are given. Moreover, the Weingarten map of an OT-ruled surface is obtained and the normal curvature, the geodesic curvature and the geodesic torsion of any curve lying on surface are obtained. Finally, some examples related to helices and slant helices are introduced.
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, respectively. The results are either not new or are minor extensions of known results, but the method, which involves introducing a different conformal structure on the surface than the one induced by the first fundamental form, is different from the one used by Hopf and requires less techn…
The fact that minimal surfaces in the four-dimensional Euclidean space admit natural parameters implies that any minimal surface is determined uniquely up to a motion by two curvature functions, satisfying a system of two PDE's (the system of natural PDE's). In fact this solves the problem of Lund-Regge for minimal surfaces. Using the corresponding result for minimal surfaces in the three-dimensional Euclidean space, we solve explicitly the system of natural PDE's, expressing any solution by virtue of two holomorphic functions in the Gauss plane. We find the relation between two pairs of holom…
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 $\R^{n+1}$.
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 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…
In this paper, we deal with the linear Weingarten factorable surfaces in the isotropic 3-space I^{3} satisfying the relation aK+bH=c, where K is the relative curvature and H the isotropic mean curvature, a,b,cR. We obtain a complete classification for such surfaces in I^{3}. As a further study, we classify all graph surfaces in I^{3} satisfying the relation K=H^{2}, which is the equality case of the famous Euler inequality for surfaces in a Euclidean space.
The study of the relation between the Weierstrass inducing formulae for constant mean curvature surfaces and the completely integrable euclidean nonlinear sigma-model suggests a connection among integrable sigma -models in a background and other type of surfaces. We show how a generalization of the Weierstrass representation can be achieved and we establish a connection with the Weingarten surfaces. We suggest also a possible generalization for two-dimensional surfaces immersed in a flat space R^8 with Euclidean metric.
Lecture notes from the mini-course "Topics in Lorentz Geometry" taught at the University of São Paulo, in March/2019. The text has three parts: (i) an overall view of linear algebra in the pseudo-Euclidean space $\mathbb{R}^n_ν$, with focus on Lorentz-Minkowski space and its role in Physics; (ii) a version of the Fundamental Theorem of Curves in 3-dimensional Lorentz-Minkowski space, with adaptations to curves with degenerate osculating plane; (iii) the problem of the diagonalization of the Weingarten map for timelike surfaces, local classification of surfaces with constant Gaussian curvature …
We show that two properly embedded self-shrinkers in Euclidean space that are sufficiently separated at infinity must intersect at a finite point. The proof is based on a localized version of the Reilly formula applied to a suitable f-harmonic function with controlled gradient. In the immersed case, a new direct proof of the generalized half-space property is also presented.
It is known that any maximal space-like surface without isotropic points in the four-dimensional pseudo-Euclidean space with neutral metric admits locally geometric parameters which are special case of isothermal parameters. With respect to such parameters the surface is determined uniquely up to a motion by the Gauss curvature and the curvature of the normal connection, which satisfy a system of two PDE's (the system of natural PDE's). For any maximal space-like surface parametrized by canonical parameters we obtain a special Weierstrass representation -- canonical Weierstrass representation.…
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…
We study the neutral Kähler metric on the space of time-like lines in Lorentzian ${\Bbb{E}}^3_1$, which we identify with the total space of the tangent bundle to the hyperbolic plane. We find all of the infinitesimal isometries of this metric, as well as the geodesics, and interpret them in terms of the Lorentzian metric on ${\Bbb{E}}^3_1$. In addition, we give a new characterisation of Weingarten surfaces in Euclidean ${\Bbb{E}}^3$ and Lorentzian ${\Bbb{E}}^3_1$ as the vanishing of the scalar curvature of the associated normal congruence in the space of oriented lines. Finally, we relate our …
We study surfaces with parallel normalized mean curvature vector field in Euclidean or Minkowski 4-space. On any such surface we introduce special isothermal parameters (canonical parameters) and describe these surfaces in terms of three invariant functions. We prove that any surface with parallel normalized mean curvature vector field parametrized by canonical parameters is determined uniquely up to a motion in Euclidean (or Minkowski) space by the three invariant functions satisfying a system of three partial differential equations. We find examples of surfaces with parallel normalized mean …
In the tangent plane at any point of a surface in the four-dimensional Euclidean space we consider an invariant linear map of Weingarten-type and find a geometrically determined 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 surfaces in the four-dimensional Euclidean space, determined by conditions on their invariants, can be in…
We construct free boundary minimal surfaces (FBMS) embedded in the unit ball in the Euclidean three-space which are compact, lie arbitrarily close to the boundary unit sphere, are of genus zero, and their boundary has an arbitrarily large number of connected boundary components. The construction is by PDE gluing methods and the surfaces are desingularizations of unions of many catenoidal annuli and two flat discs. The union of the boundaries of the catenoidal annuli and discs is the union of a large finite number of parallel circles contained in the unit sphere, with each parallel circle conta…
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 introduce a class of potential submanifolds in pseudo-Euclidean spaces (each N-dimensional potential submanifold is a special flat torsionless submanifold in a 2N-dimensional pseudo-Euclidean space) and prove that each N-dimensional Frobenius manifold can be locally represented as an N-dimensional potential submanifold. We show that all potential submanifolds bear natural special structures of Frobenius algebras on their tangent spaces. These special Frobenius structures are generated by the corresponding flat first fundamental form and the set of the second fundamental forms of the submani…
We review classical approaches to the problem of isometrically embedding a Riemannian surface into Euclidean 3-space, including coordinate-based approaches exposited by Darboux and Eisenhart, as well as the moving-frames based approaches advocated by Cartan. In particular, the first approach involves reducing the problem to solving a single PDE; settling the question of when this PDE is integrable by the method of Darboux is the subject of this short note. This is surprisingly easy, since it is related to the analogous question for the isometric embedding system arising from the moving frames …
A twisted-austere $k$-fold $(M, μ)$ in $\mathbb R^n$ consists of a $k$-dimensional submanifold $M$ of $\mathbb R^n$ together with a closed $1$-form $μ$ on $M$ such that the `twisted conormal bundle' $N^* M + μ$ is a special Lagrangian submanifold of $\mathbb C^n$. The 1-form $μ$ and the second fundamental form of $M$ must satisfy a particular system of coupled nonlinear second order PDE. We first review these twisted-austere conditions and give an explicit example. Then we focus on twisted-austere 3-folds, giving a geometric description of all solutions when the base $M$ is a cylinder and when…
We prove that the associativity equations of two-dimensional topological quantum field theories are very natural reductions of the fundamental nonlinear equations of the theory of submanifolds in pseudo-Euclidean spaces and give a natural class of potential flat torsionless submanifolds. We show that all potential flat torsionless submanifolds in pseudo-Euclidean spaces bear natural structures of Frobenius algebras on their tangent spaces. These Frobenius structures are generated by the corresponding flat first fundamental form and the set of the second fundamental forms of the submanifolds (i…
This paper is concerned with the Dirichlet eigenvalue problem associated to the $\infty$-Laplacian in metric spaces. We establish a direct PDE approach to find the principal eigenvalue and eigenfunctions in a proper geodesic space without assuming any measure structure. We provide an appropriate notion of solutions to the $\infty$-eigenvalue problem and show the existence of solutions by adapting Perron's method. Our method is different from the standard limit process via the variational eigenvalue formulation for $p$-Laplacian in the Euclidean space.
