Families Of Such Surfaces Research Articles

Wave maps and constant curvature surfaces: singularities and bifurcations

Wave maps (or Lorentzian-harmonic maps) from a $1+1$-dimensional Lorentz space into the $2$-sphere are associated to constant negative Gaussian curvature surfaces in Euclidean 3-space via the Gauss map, which is harmonic with respect to the metric induced by the second fundamental form. We give a method for constructing germs of Lorentzian-harmonic maps from their $k$-jets and use this construction to study the singularities of such maps. We also show how to construct pseudospherical surfaces with prescribed singularities using loop groups. We study the singularities of pseudospherical surfaces and obtain their bifurcations in generic 1-parameter families of such surfaces.

Principal Foliations of Surfaces near Ellipsoids

The lines of curvature of a surface embedded in $\R^3$ comprise its principal foliations. Principal foliations of surfaces embedded in $\R^3$ resemble phase portraits of two dimensional vector fields, but there are significant differences in their geometry because principal foliations are not orientable. The Poincar\'e-Bendixson Theorem precludes flows on the two sphere $S^2$ with recurrent trajectories larger than a periodic orbit, but there are convex surfaces whose principal foliations are closely related to non-vanishing vector fields on the torus $T^2$. This paper investigates families of such surfaces that have dense lines of curvature at a Cantor set $C$ of parameters. It introduces discrete one dimensional return maps of a cross-section whose trajectories are the intersections of a line of curvature with the cross-section. The main result proved here is that the return map of a generic surface has \emph{breaks}; i.e., jump discontinuities of its derivative. Khanin and Vul discovered a qualitative difference between one parameter families of smooth diffeomorphisms of the circle and those with breaks: smooth families have positive Lebesgue measure sets of parameters with irrational rotation number and dense trajectories while families of diffeomorphisms with a single break do not. This paper discusses whether Lebesgue almost all parameters yield closed lines of curvature in families of embedded surfaces.

Extensions maximales et classification des tores lorentziens munis d’un champ de Killing

We study the simply connected inextendable Lorentzian surfaces admitting a Killing vector field. We construct a natural family of such surfaces, that we call extensions. They are characterized by a condition of symmetry, the reflexivity, and a by a rather weak completeness assumption, the absence of saddles at infinity. Considering these surfaces as model spaces, we study their minimal quotients, divisible open sets and conjugate points. We show uniformisation results (by an open subset of one of these universal extensions, which is uniquely determined) in the following cases: compact surfaces and analytical surfaces. It allows us to give a classification of Lorentzian tori and Klein bottles with a Killing vector field.

Complete biconservative surfaces in the hyperbolic space [formula omitted

We construct simply connected, complete, non-CMC biconservative surfaces in the 3-dimensional hyperbolic space H3, in an intrinsic and extrinsic way. We obtain three families of such surfaces, and, for each surface, the set of points where the gradient of the mean curvature function does not vanish is dense and has two connected components. In the intrinsic approach, we first construct a simply connected, complete abstract surface and then prove that it admits a unique biconservative immersion in H3. Working extrinsically, we use the images of the explicit parametric equations and a gluing process to obtain our surfaces. They are made up of circles (or hyperbolas, or parabolas, respectively) which lie in 2-affine parallel planes and touch a certain curve lying in a totally geodesic hyperbolic surface H2 in H3.

Further Deformations of the Axially-Symmetric Non-Bending Surfaces

We consider a special class of symmetrically loaded thin shells of revolution, which in the presence of certain disturbances of the equilibrium deform without bending. The whole family of such surfaces can be regarded as complementary set to the four classes of equilibrium states of non-bending surfaces that we have studied in some depth recently. The new surfaces (with a few exceptions) have not a closed form description in elementary functions. Here we present their explicit parameterizations in terms of elliptic integrals.

Algebraic surfaces with pg = q = 1,K2 = 4 and nonhyperelliptic Albanese fibrations of genus 4

In this article, we study minimal algebraic surfaces with and nonhyperelliptic Albanese fibrations of genus 4. We construct for the first time a family of such surfaces as complete intersections of type (2, 3) in a -bundle over an elliptic curve. For the surfaces we construct here, the direct image of the canonical sheaf under the Albanese map is decomposable (which is a topological invariant property). Moreover, we prove that all minimal surfaces with and nonhyperelliptic Albanese fibrations of genus 4 such that the direct image of the canonical sheaf under the Albanese map is decomposable are contained in our family. As a consequence, we show that these surfaces constitute a four-dimensional irreducible subset of , the Gieseker moduli space of minimal surfaces with . Moreover, the closure of is an irreducible component of .

Families of spherical surfaces and harmonic maps

We study singularities of constant positive Gaussian curvature surfaces and determine the way they bifurcate in generic 1-parameter families of such surfaces. We construct the bifurcations explicitly using loop group methods. Constant Gaussian curvature surfaces correspond to harmonic maps, and we examine the relationship between the two types of maps and their singularities. Finally, we determine which finitely $$\mathcal {A}$$ -determined map-germs from the plane to the plane can be represented by harmonic maps.

On Bäcklund and Ribaucour transformations for hyperbolic linear Weingarten surfaces

We consider Bäcklund transformations for hyperbolic linear Weingarten surfaces in Euclidean 3-space. The composition of these transformations is obtained in the Permutability Theorem that generates a 4-parameter family of surfaces of the same type. Since a Ribaucour transformation of a hyperbolic linear Weingarten surface also gives a 4-parameter family of such surfaces, one has the following natural question. Are these two methods equivalent, as it occurs with surfaces of constant positive Gaussian curvature or constant mean curvature? By obtaining necessary and sucient conditions for the surfaces given by the two procedures to be congruent.The analytic interpretation of the geometric results is given in terms of solutions of the sine-Gordon equation.

Zero mean curvature surfaces in Lorentz--Minkowski 3-space which change type across a light-like line

It is well-known that space-like maximal surfaces and time-like minimal surfaces in Lorentz--Minkowski $3$-space $\boldsymbol{R}^{3}_{1}$ have singularities in general. They are both characterized as zero mean curvature surfaces. We are interested in the case where the singular set consists of a light-like line, since this case has not been analyzed before. As a continuation of a previous work by the authors, we give the first example of a family of such surfaces which change type across a light-like line. As a corollary, we also obtain a family of zero mean curvature hypersurfaces in $\boldsymbol{R}^{n+1}_{1}$ that change type across an ($n-1$)-dimensional light-like plane.

On the Singularities of Surfaces Ruled by Conics

We classify the singularities of a surface ruled by conics: they are rational double points of type A n or D n . This is proved by showing that they arise from a precise series of blow-ups of a suitable surface geometrically ruled by conics. We determine also the family of such surfaces which are birational models of a given surface ruled by conics and obtained in a “minimal way” from it.

On the uniqueness of elliptic K3 surfaces with maximal singular fibre

We explicitly determine the elliptic K3 surfaces with a maximal singular fibre. If the characteristic of the ground field is different from 2, for each of the two possible maximal fibre types, I19 and I ∗ 14, the surface is unique. In characteristic 2 the maximal fibre types are I18 and I ∗ 13, and there exist two (resp. one) one-parameter families of such surfaces.

On recurrence and ergodicity for geodesic flows on non-compact periodic polygonal surfaces

AbstractWe study billiard dynamics on non-compact polygonal surfaces with a free, cocompact action of ℤ or ℤ2. In the ℤ-periodic case, we establish criteria for conservativity. In the ℤ2-periodic case, we study a particular family of such surfaces, the rectangular Lorenz gas. Assuming that the obstacles are sufficiently small, we obtain the ergodic decomposition of directional billiards for a finite but asymptotically dense set of directions. This is based on our study of ergodicity for ℤd-valued cocycles over irrational rotations.

Complete classification of parallel Lorentz surfaces in four-dimensional neutral pseudosphere

A Lorentz surface of an indefinite space form is called parallel if its second fundamental form is parallel with respect to the Van der Waerden–Bortolotti connection. Such surfaces are locally invariant under the reflection with respect to the normal space at each point. Parallel surfaces are important in geometry as well as in general relativity since extrinsic invariants of such surfaces do not change from point to point. Parallel Lorentz surfaces in four-dimensional (4D) Lorentzian space forms are classified by Chen and Van der Veken [“Complete classification of parallel surfaces in 4-dimensional Lorentz space forms,” Tohoku Math. J. 61, 1 (2009)]. Recently, explicit classification of parallel Lorentz surfaces in the pseudo-Euclidean 4-space E24 and in the pseudohyperbolic 4-space H24(−1) are obtained recently by Chen et al. [“Complete classification of parallel Lorentzian surfaces in Lorentzian complex space forms,” Int. J. Math. 21, 665 (2010); “Complete classification of parallel Lorentz surfaces in neutral pseudo hyperbolic 4-space,” Cent. Eur. J. Math. 8, 706 (2010)], respectively. In this article, we completely classify the remaining case; namely, parallel Lorentz surfaces in 4D neutral pseudosphere S24(1). Our result states that there are 24 families of such surfaces in S24(1). Conversely, every parallel Lorentz surface in S24(1) is obtained from one of the 24 families. The main result indicates that there are major differences between Lorentz surfaces in the de Sitter 4-space dS4 and in the neutral pseudo 4-sphere S24.

Classification of Lorentzian surfaces with parallel mean curvature vector in % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXguY9 % gCGievaerbd9wDYLwzYbWexLMBbXgBcf2CPn2qVrwzqf2zLnharyav % P1wzZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC % 0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yq % aqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabe % qaamaaeaqbaaGcbaWefv3ySLgznfgDOjdarCqr1ngBPrginfgDObcv % 39gaiyaacqWFecFrdaqhaaWcbaGaeGOmaidabaGaeGinaqdaaaaa!49CF! $$ \mathbb{E}_2^4 $$

We completely classify Lorentzian surfaces with parallel mean curvature vector in \( \mathbb{E}_2^4 \). The main result states that there exist 22 families of such surfaces.

Stationary rotating surfaces in Euclidean space

A stationary rotating surface is a compact surface in Euclidean space whose mean curvature H at each point x satisfies 2H(x) = a r(x)2 + b, where r(x) denotes the distance from x to a fixed straight-line L, and a and b are constants. These surfaces are solutions of a variational problem that describes the shape of a drop of incompressible fluid in equilibrium by the action of surface tension when it rotates about L with constant angular velocity. The effect of gravity is neglected. In this paper we study the geometric configurations of such surfaces, focusing the relationship between the geometry of the surface and the one of its boundary. As special cases, we will consider two families of such surfaces: axisymmetric surfaces and embedded surfaces with planar boundary.

Classification of spatial surfaces with parallel mean curvature vector in pseudo-Euclidean spaces of arbitrary dimension

Spatial surfaces with parallel mean curvature vector play some important roles in general relativity, theory of harmonic maps, as well as in differential geometry. Recently, spatial surfaces in four-dimensional Minkowski space time with parallel mean curvature vector were classified by Chen and Van der Veken [“Complete classification of parallel surfaces in 4-dimensional Lorentzian space forms,” Tohoku Math. J. 61, 1 (2009)]. In this article, we completely classify spatial surfaces with parallel mean curvature vector in pseudo-Euclidean spaces of arbitrary dimension. The main result states that there exist 16 families of such surfaces. As by-product, we achieve the complete classification of spatial surfaces with parallel mean curvature vector in Minkowski space times of arbitrary dimension.

Topologically unique maximal elementary abelian group actions on compact oriented surfaces

We determine all finite maximal elementary abelian group actions on compact oriented surfaces of genus σ ≥ 2 which are unique up to topological equivalence. For certain special classes of such actions, we determine group extensions which also define unique actions. In addition, we explore in detail one of the families of such surfaces considered as compact Riemann surfaces and tackle the classical problem of constructing defining equations.

Harmonic maps and constant mean curvature surfaces in H 2 x R

We introduce a hyperbolic Gauss map into the Poincar´e disk for any surface in H2×R with regular vertical projection, and prove that if the surface has constant mean curvature H = 1/2, this hyperbolic Gauss map is harmonic. Conversely, we show that every nowhere conformal harmonic map from an open simply connected Riemann surface Σ into the Poincaré disk is the hyperbolic Gauss map of a two-parameter family of such surfaces. As an application we obtain that any holomorphic quadratic differential on Σ can be realized as the Abresch-Rosenberg holomorphic differential of some, and generically infinitely many, complete surfaces with H = 1/2 in H2 x R. A similar result applies to minimal surfaces in the Heisenberg group Nil3. Finally, we classify all complete minimal vertical graphs in H2 × R.

Classification of Totally Geodesic Surfaces in the Manifold of Directions in Physical Space

The Grassmannian of bivectors over the pseudo-Euclidean Minkowski 4-space is considered and its two-dimensional totally geodesic submanifolds are classified. The family of such surfaces is described in the language of the affine geometry of three-space. Bibliography: 5 titles.

Hypersurfaces in the non-flat Lorentzian space forms with a characteristic eigenvector field

In a series of early papers, with the aim of knowing of the shape of a pseudo-Riemannian hypersurface satisfying a certain differential equation in the induced Laplacian, we found a remarkable family of hypersurfaces in the Lorentz-Minkowski space whose mean curvature vector is an eigenvector of the Laplacian. Actually, the last two authors showed in [8] that the equation ∆H = λH , for a real constant λ, characterizes the family of surfaces in L3 made up of the quite interesting B-scrolls and the so-called standard examples, as well as minimal surfaces. Looking at those results obtained for surfaces in L3, the following geometric question was stated in [9] for hypersurfaces in Ln+1 (n > 2): Does the equation ∆H = λH mean that both the mean and the scalar curvatures of the hypersurface are constant? We were able to give a partial solution to that problem, since we had needed to do an additional hypothesis on the degree of the minimal polynomial of the shape operator. It is worth pointing out that the additional assumption was mainly made to control the position vector field of the hypersurface into Ln+1. Now when the ambient space is a non-flat pseudoRiemannian space form, Sn+1 ν (r) or Hn+1 ν (r), then the hypersurface is of codimension two in Rn+2 ν or R ν+1 , respectively, but Sn+1 ν (r) and Hn+1 ν (r) being both totally umbilical hypersurfaces in the corresponding pseudo-Euclidean space, it seems reasonable to hope for a richer classification of hypersurfaces into those spaces by means of the equation ∆H = λH . Or even, one looks for getting a complete answer to the stated problem in non-flat ambient spaces. In this paper we give a classification of surfaces in the 3-dimensional non-flat Lorentzian space forms satisfying the equation ∆H = λH . We show that the family of such surfaces consists of minimal, totally umbilical and B-scroll surfaces. As for hypersurfaces we suppose that their shape operators have no complex eigenvalues. This condition does not seem as restrictive as one could think, in view of examples and results given in section 5. Actually, we find that family is set up by minimal, totally umbilical and so-called generalized umbilical hypersurfaces, which are nothing but a natural generalization of B-scrolls.