Constant Curvature Surfaces Research Articles

On the classification of Killing submersions and their isometries

A Killing submersion is a Riemannian submersion from an orientable 3-manifold to an orientable surface whose fibers are the integral curves of a unit Killing vector field in the 3-manifold. We classify all Killing submersions over simply-connected Riemannian surfaces and give explicit models for many Killing submersions including those over simply-connected constant Gaussian curvature surfaces. We also fully describe the isometries of the total space preserving the vertical direction. As a consequence, we prove that the only simply-connected homogeneous 3-manifolds which admit a structure of Killing submersion are the E(\kappa,\tau)-spaces, whose isometry group has dimension at least 4.

Constant Gaussian curvature surfaces in the 3-sphere via loop groups

In this paper we study constant positive Gauss curvature $K$ surfaces in the 3-sphere $S^3$ with $0<K<1$ as well as constant negative curvature surfaces. We show that the so-called normal Gauss map for a surface in $S^3$ with Gauss curvature $K<1$ is Lorentz harmonic with respect to the metric induced by the second fundamental form if and only if $K$ is constant. We give a uniform loop group formulation for all such surfaces with $K\neq 0$, and use the generalized d'Alembert method to construct examples. This representation gives a natural correspondence between such surfaces with $K<0$ and those with $0<K<1$.

Quantum field theory in curved graphene spacetimes, Lobachevsky geometry, Weyl symmetry, Hawking effect, and all that

The solutions of many issues, of the ongoing efforts to make deformed graphene a tabletop quantum field theory in curved spacetimes, are presented. A detailed explanation of the special features of curved spacetimes, originating from embedding portions of the Lobachevsky plane into $\mathbf{R}^3$, is given, and the special role of coordinates for the physical realizations in graphene, is explicitly shown, in general, and for various examples. The Rindler spacetime is reobtained, with new important differences with respect to earlier results. The de Sitter spacetime naturally emerges, for the first time, paving the way to future applications in cosmology. The role of the BTZ black hole is also briefly addressed. The singular boundary of the pseudospheres, "Hilbert horizon", is seen to be closely related to event horizon of the Rindler, de Sitter, and BTZ kind. This gives new, and stronger, arguments for the Hawking phenomenon to take place. An important geometric parameter, $c$, overlooked in earlier work, takes here its place for physical applications, and it is shown to be related to graphene's lattice spacing, $\ell$. It is shown that all surfaces of constant negative curvature, ${\cal K} = -r^{-2}$, are unified, in the limit $c/r \to 0$, where they are locally applicable to the Beltrami pseudosphere. This, and $c = \ell$, allow us a) to have a phenomenological control on the reaching of the horizon; b) to use spacetimes different than Rindler for the Hawking phenomenon; c) to approach the generic surface of the family. An improved expression for the thermal LDOS is obtained. A non-thermal term for the total LDOS is found. It takes into account: a) the peculiarities of the graphene-based Rindler spacetime; b) the finiteness of a laboratory surface; c) the optimal use of the Minkowski quantum vacuum, through the choice of this Minkowski-static boundary.

Translation surfaces of constant curvatures in a simply isotropic space

In this paper we describe, up to a congruence, translation surfaces in a simply isotropic space having constant isotropic Gaussian or mean curvature. It turns out that, contrary to the Euclidean case, there exist translation surfaces with constant Gaussian curvature \(K\ne 0\) and translation surfaces with constant mean curvature \(H\ne 0\) that are not cylindrical. Furthermore, we investigate a class of Weingarten translation surfaces.

Eulerian relative equilibria of the curved 3-body problems in 𝐒²

We consider the gravitational motion of n n point particles with masses m 1 m_1 , m 2 m_2 , … \ldots , m n &gt; 0 m_n&gt;0 on surfaces of constant Gaussian curvature. Based on the work of Diacu and his co-authors, we derive the law of universal gravitation in spaces of constant curvature. Using the results, we examine all possible 3 3 -body configurations that can generate geodesic relative equilibria. We prove the existence of all acute triangle Eulerian relative equilibria and get a necessary and sufficient condition for the existence of obtuse triangle Eulerian relative equilibria. We also show that any three positive masses can generate Eulerian relative equilibria.

Turbulence on Hyperbolic Plane: The Fate of Inverse Cascade

We describe ideal incompressible hydrodynamics on the hyperbolic plane which is an infinite surface of constant negative curvature. We derive equations of motion, general symmetries and conservation laws, and then consider turbulence with the energy density linearly increasing with time due to action of small-scale forcing. In a flat space, such energy growth is due to an inverse cascade, which builds a constant part of the velocity autocorrelation function proportional to time and expanding in scales, while the moments of the velocity difference saturate during a time depending on the distance. For the curved space, we analyze the long-time long-distance scaling limit, that lives in a degenerate conical geometry, and find that the energy-containing mode linearly growing with time is not constant in space. The shape of the velocity correlation function indicates that the energy builds up in vortical rings of arbitrary diameter but of width comparable to the curvature radius of the hyperbolic plane. The energy current across scales does not increase linearly with the scale, as in a flat space, but reaches a maximum around the curvature radius. That means that the energy flux through scales decreases at larger scales so that the energy is transferred in a non-cascade way, that is the inverse cascade spills over to all larger scales where the energy pumped into the system is cumulated in the rings. The time-saturated part of the spectral density of velocity fluctuations contains a finite energy per unit area, unlike in the flat space where the time-saturated spectrum behaves as k^{-5/3}.

Hyperbolic Plateau problems

We consider surfaces of constant Gaussian curvature immersed in \(3\)-dimensional manifolds, and we strengthen the compactness result of Labourie in the case where the ambient manifold is \(3\)-dimensional hyperbolic space. This allows us to prove results of existence of solutions to the asymptotic Plateau problem, as defined by Labourie, and the continuous dependence of these solutions on the data.

Limit Theorems for Horocycle Flows

The main results of this paper are limit theorems for horocycle flows on compact surfaces of constant negative curvature. One of the main objects of the paper is a special family of horocycle-invariant finitely additive Holder measures on rectifiable arcs. An asymptotic formula for ergodic integrals for horocycle flows is obtained in terms of the finitely-additive measures, and limit theorems follow as a corollary of the asymptotic formula. The objects and results of this paper are similar to those in [15], [16], [4] and [5] for translation flows on flat surfaces. The arguments are based on the classification of invariant distributions for horocycle flows established in [12].

Maximum principles and isoperimetric inequalities for some Monge–Ampère-type problems

In this note we derive a maximum principle for an appropriate functional combination of u(x) and |∇u|2, where u(x) is a strictly convex classical solution to a general class of Monge–Ampère equations. This maximum principle is then employed to establish some isoperimetric inequalities of interest in the theory of surfaces of constant Gauss curvature in RN+1.

Maximum and minimum principles for a class of Monge-Ampère equations in the plane, with applications to surfaces of constant Gauss curvature

In this article we investigate a general class of Monge-Ampere equations in the plane, including the constant Gauss curvature equation. Our first aim is to prove some maximum and minimum principles for suitable $P$-functions, in the sense of L.E. Payne. Then, these new principles are employed to solve a general class of overdetermined Monge-Ampere problems and to investigate two boundary value problems for the constant Gauss curvature equation. More precisely, when the constant Gauss curvature equation is subject to the homogeneous Dirichlet boundary condition, we prove several isoperimetric inequalities, while when it is subject to the contact angle boundary condition, some necessary conditions of solvability, involving the curvature of the boundary of the underlying domain and the given contact angle, are derived.

Contact of ropes and orthotropic rough surfaces

Contact between arbitrary curved ropes and arbitrary curved rough orthotropic surfaces has been revised from the geometrical point of view. Variational equations for the equilibrium of ropes on orthotropic rough surfaces are derived, first, using the consistent variational inclusion of frictional contact constraints via Karush‐Kuhn‐Tucker conditions expressed in Darboux basis. Then, the systems of differential equations are derived for both statics and dynamics of ropes on a rough surface depending on the sticking‐sliding condition for orthotropic Coulomb's friction. Three criteria are found to be fulfilled during the static equilibrium of a rope on a rough surface: “no separation”, condition for dragging coefficient of friction and inequality for tangential forces at the end of the rope. The limit tangential loads still preserve the famous “Euler view” T = T0e± ω s for the curves and surfaces of constant curvature. It is shown that the curve of the maximum tension of a rough orthotropic surface is geodesic. Equations of motion are derived in the case if the sliding criteria is fulfilled and there is “no separation”. Various cases possessing analytical solutions of the derived system, including Euler case and a spiral rope on a cylinder are shown as examples of application of the derived theory.

The Minkowski problem, new constant curvature surfaces in ℝ3, and some applications

Abstract Let 𝔪 ∈ ℕ ${\mathfrak {m}\in \mathbb {N}}$ , 𝔪 ≥ 2 ${\mathfrak {m}\ge 2}$ , and let { p j } j = 1 𝔪 ${\lbrace p_j\rbrace _{j=1}^\mathfrak {m}}$ be a finite subset of 𝕊2 such that 0 → ∈ ℝ 3 ${\vec{0}\in \mathbb {R}^3}$ lies in its positive convex hull. In this paper we make use of the classical Minkowski problem, to show the complete family of smooth convex bodies 𝒦 in ℝ3 whose boundary surface consists of an open surface S with constant Gauss curvature (respectively, constant mean curvature) and 𝔪 planar compact discs D ¯ 1 , ... , D ¯ 𝔪 ${\overline{D}_1,\ldots ,\overline{D}_\mathfrak {m}}$ , such that the Gauss map of S is a homeomorphism onto 𝕊 2 - { p j } j = 1 𝔪 ${\mathbb {S}^2-\lbrace p_j\rbrace _{j=1}^\mathfrak {m}}$ and D j ⊥ p j ${D_j\bot p_j}$ , for all j . ${j.}$ We derive applications to existence of harmonic diffeomorphisms between domains of 𝕊2, existence of capillary surfaces in ℝ3, and a Hessian equation of Monge–Ampère type.

Phyllotaxis on surfaces of constant Gaussian curvature

A close-packed organization with circular symmetry of a large number of small discs on a plane is obtained when the centres of the discs are distributed according to the algorithm of phyllotaxis. We study here the distributions obtained on surfaces of constant Gaussian curvatures, positive for the sphere or negative for the hyperbolic plane. We examine how the properties of homogeneity, isotropy and self-similarity, typical of the distribution on the plane and resulting from the presence of circular grain boundaries with quasicrystalline sequences, are affected by the curvature of the bearing surface. The quasicrystalline sequences of the grain boundaries appear indeed to be structural invariants, but the widths of the grains they separate vary differently with the curvature of the surface. The self-similarity of the whole organization observed on the plane is therefore lost on the hyperbolic plane and the sphere. The evolutions of the local order within the grains show no differences except on the equatorial belt of the sphere where the isotropy is decreased owing to the symmetry of this finite surface around its equator.

Magnetic fields in 2D and 3D sphere

In this note we study the Landau–Hall problem in the 2D and 3D unit sphere, that is, the motion of a charged particle in the presence of a static magnetic field. The magnetic flow is completely determined for any Riemannian surface of constant Gauss curvature, in particular, the unit 2D sphere. For the 3D case we consider Killing magnetic fields on the unit sphere, and we show that the magnetic flowlines are helices with the given Killing vector field as its axis.

Graphene: QFT in curved spacetimes close to experiments

A recently proposed step-by-step procedure, to merge the low-energy physics of the π-bonds electrons of graphene, and quantum field theory on curved spacetimes, is recalled. The last step there is the proposal of an experiment to test a Hawking-Unruh effect, emerging from the model, that manifests itself as an exact (within the model) prediction for the electronic local density of states, in the ideal case of the graphene membrane shaped as a Beltrami pseudosphere. A discussion about one particular attempt to experimentally test the model on molecular graphene is presented, and it is taken as an excuse to solve some basic issues that will help future experiments. In particular, it is stated that the effect should be visible on generic surfaces of constant negative Gaussian curvature, that are infinite in number.

Paper Surface Geometry: Surveying a Locally Euclidean Universe

The concepts of parallel transport and intrinsic (Gaussian) curvature arising in the differential geometry of surfaces may be pleasantly and concretely investigated using paper models and familiar notions of length and angle. This paper introduces parallel transport and curvature in the context of “locally Euclidean” surfaces: polyhedra, for which curvature is concentrated at isolated points, and “polycones”, for which curvature is concentrated along circular arcs. The geometry of polycones is used to recover a strikingly simple intrinsic formula for the Gaussian curvature of a surface of rotation. We give instructions for building paper models of the catenoid and surfaces of constant Gaussian curvature.

Surfaces of constant curvature in [formula omitted] with isolated singularities

We prove that finite area isolated singularities of surfaces with constant positive curvature K>0 in R3 are removable singularities, branch points or immersed conical singularities. We describe the space of immersed conical singularities of such surfaces in terms of the class of real analytic closed locally strictly convex curves in S2 with admissible cusp singularities, characterizing when the singularity is actually embedded. In the global setting, we describe the space of peaked spheres in R3, i.e. compact convex surfaces of constant curvature K>0 with a finite number of singularities, and give applications to harmonic maps and constant mean curvature surfaces.

Structures Of Surfaces With Non-Zero Constant Curvature

In this paper, surfaces of positive constant curvature and those of negative curvature are studied. In certain cases, a very few important structures of this class of surfaces have been obtained. DOI: http://dx.doi.org/10.3329/ganit.v32i0.13644 GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 32 (2012) 37-41

The geometric Cauchy problem for surfaces with Lorentzian harmonic Gauss maps

The geometric Cauchy problem for a class of surfaces in a pseudo-Riemannian manifold of dimension 3 is to find the surface which contains a given curve with a prescribed tangent bundle along the curve. We consider this problem for constant negative Gauss curvature surfaces (pseudospherical surfaces) in Euclidean 3-space, and for timelike constant non-zero mean curvature (CMC) surfaces in the Lorentz-Minkowski 3- space. We prove that there is a unique solution if the prescribed curve is non-characteristic, and for characteristic initial curves (asymptotic curves for pseudospherical surfaces and null curves for timelike CMC) it is nec- essary and sufficient for similar data to be prescribed along an additional characteristic curve that intersects the first. The proofs also give a means of constructing all solutions using loop group techniques. The method used is the infinite dimensional d'Alembert type representation for surfaces as- sociated with Lorentzian harmonic maps (1-1 wave maps) into symmetric spaces, developed since the 1990's. Explicit formulae for the potentials in terms of the prescribed data are given, and some applications are consid- ered.

Backlund’s Theorem for Spacelike Surfacesin Minkowski 3-Space

By using the method of moving frames the Backlund's theorem and its applicationfor spacelike surfaces is introduced. The results leads to correspondence between the solutions of the Sine-Gordon equation and spacelike surfaces of constant positive Gaussian curvatures.