Constant Curvature Surfaces Research Articles

Compact Surfaces with Constant Gaussian Curvature in Product Spaces

We prove that the only compact surfaces of positive constant Gaussian curvature in \({\mathbb{H}^{2}\times\mathbb{R}}\) (resp. positive constant Gaussian curvature greater than 1 in \({\mathbb{S}^{2}\times\mathbb{R}}\)) whose boundary Γ is contained in a slice of the ambient space and such that the surface intersects this slice at a constant angle along Γ, are the pieces of a rotational complete surface. We also obtain some area estimates for surfaces of positive constant Gaussian curvature in \({\mathbb{H}^{2}\times\mathbb{R}}\) and positive constant Gaussian curvature greater than 1 in \({\mathbb{S}^{2}\times\mathbb{R}}\) whose boundary is contained in a slice of the ambient space. These estimates are optimal in the sense that if the bounds are attained, the surface is again a piece of a rotational complete surface.

On the Gauss curvature of compact surfaces in homogeneous 3-manifolds

Compact flat surfaces of homogeneous Riemannian 3-manifolds with isometry group of dimension 4 are classified. Nonexistence results for compact constant Gauss curvature surfaces in these 3-manifolds are established.

A minimal immersion of the hyperbolic plane into the neutral pseudo-hyperbolic 4-space and its characterization

It is well known that there are no minimal surfaces of constant curvature lying fully in the hyperbolic 4-space H 4(−1). In contrast, in this article we discover a minimal immersion of the hyperbolic plane \({H^2(-\frac{1}{3})}\) of curvature \({-\frac{1}{3}}\) into the neutral pseudo-hyperbolic 4-space \({H^4_2(-1)}\). Moreover, we prove that, up to rigid motions of \({H^4_2(-1)}\), this minimal immersion provides the only space-like parallel minimal surface lying fully in \({H^4_2(-1)}\).

A curvature theory for discrete surfaces based on mesh parallelity

We consider a general theory of curvatures of discrete surfaces equipped with edgewise parallel Gauss images, and where mean and Gaussian curvatures of faces are derived from the faces’ areas and mixed areas. Remarkably these notions are capable of unifying notable previously defined classes of surfaces, such as discrete isothermic minimal surfaces and surfaces of constant mean curvature. We discuss various types of natural Gauss images, the existence of principal curvatures, constant curvature surfaces, Christoffel duality, Koenigs nets, contact element nets, s-isothermic nets, and interesting special cases such as discrete Delaunay surfaces derived from elliptic billiards.

Minimal surfaces in pseudohermitian geometry

We consider surfaces immersed in three-dimensional pseudohermitian manifolds. We define the notion of (p-)mean curvature and of the associated (p-)minimal surfaces, extending some concepts previously given for the (flat) Heisenberg group. We interpret the p-mean curvature not only as the tangential sublaplacian of a defining function, but also as the curvature of a characteristic curve, and as a quantity in terms of calibration geometry. As a differential equation, the p-minimal surface equation is degenerate (hyperbolic and elliptic). To analyze the singular set (i.e., the set where the (p-)area integrand vanishes), we formulate some extension theorems, which describe how the characteristic curves meet the singular set. This allows us to classify the entire solutions to this equation and to solve a Bernstein-type problem (for graphs over the xy-plane) in the Heisenberg group H1. In H1, identified with the Euclidean space R3, the p-minimal surfaces are classical ruled surfaces with the rulings generated by Legendrian lines. We also prove a uniqueness theorem for the Dirichlet problem under a condition on the size of the singular set in two dimensions, and generalize to higher dimensions without any size control condition. We also show that there are no closed, connected, C2 smoothly immersed constant p-mean curvature or p-minimal surfaces of genus greater than one in the standard S3. This fact continues to hold when S3 is replaced by a general pseudohermitian 3-manifold. Mathematics Subject Classification (2000): 35L80 (primary); 35J70, 32V20, 53A10, 49Q10 (secondary).

Bianchi–Bäcklund transforms and dressing actions, revisited

We characterize Bianchi–Backlund transformations of surfaces of positive constant Gauss curvature in terms of dressing actions of the simplest type on the space of harmonic maps.

Classification of Lagrangian surfaces of curvature ɛ in non-flat Lorentzian complex space form % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqr1ngBPrgifHhDYfgasaacH8WrFz0dbbf9q8 % WrFfeuY-Hhbbf9v8qqaqFr0dd9qqFj0dXdbba91qpepGe9FjuP0-is % 0dXdbba9pGe9xq-Jbba9suk9fr-xfr-xfrpeWZqaaeaabiGaciaaca % qabeaadaabauaaaOqaaiqbd2eanzaaiaWaa0baaSqaaiabigdaXaqa % aiabikdaYaaakiabcIcaOiabisda0eXafv3ySLgzGmvETj2BSbacgi % Gae8xTduMaeiykaKcaaa!4CAA! $$ \tilde M_1^2 (4\varepsilon ) $$ )

It is well known that a totally geodesic Lagrangian surface in a Lorentzian complex space form \( \tilde M_1^2 (4\varepsilon ) \) of constant holomorphic sectional curvature 4ɛ is of constant curvature ɛ. A natural question is “Besides totally geodesic ones how many Lagrangian surfaces of constant curvature ɛ in \( \tilde M_1^2 (4\varepsilon ) \) are there?” In an earlier paper an answer to this question was obtained for the case ɛ = 0 by Chen and Fastenakels. In this paper we provide the answer to this question for the case ɛ ≠ 0. Our main result states that there exist thirty-five families of Lagrangian surfaces of curvature ɛ in \( \tilde M_1^2 (4\varepsilon ) \) with ɛ ≠ 0. Conversely, every Lagrangian surface of curvature ɛ ≠ 0 in \( \tilde M_1^2 (4\varepsilon ) \) is locally congruent to one of the Lagrangian surfaces given by the thirty-five families.

Hamiltonian formulation of surfaces with constant Gaussian curvature

Dirac's method for constrained Hamiltonian systems is used to describe surfaces of constant Gaussian curvature. A geometrical free energy, for which these surfaces are equilibrium states, is introduced and interpreted as an action. An equilibrium surface can then be generated by the evolution of a closed space curve. Since the underlying action depends on second derivatives, the velocity of the curve and its conjugate momentum must be included in the set of phase-space variables. Furthermore, the action is linear in the acceleration of the curve and possesses a local symmetry—reparametrization invariance—which implies primary constraints in the canonical formalism. These constraints are incorporated into the Hamiltonian through Lagrange multiplier functions that are identified as the components of the acceleration of the curve. The formulation leads to four first-order partial differential equations, one for each canonical variable. With the appropriate choice of parametrization, only one of these equations has to be solved to obtain the surface which is swept out by the evolving space curve. To illustrate the formalism, several evolutions of pseudospherical surfaces are discussed.

Growth rates for geometric complexities and counting functions in polygonal billiards

AbstractWe introduce a new method for estimating the growth of various quantities arising in dynamical systems. Applying our method to polygonal billiards on surfaces of constant curvature, we obtain estimates almost everywhere on direction complexities and position complexities. As a byproduct, we recover the power bounds of Boshernitzan on the number of billiard orbits between almost all pairs of points in a planar polygon [M. Boshernitzan. Private letter to H. Masur (1986)].

Green’s Function for the Hodge Laplacian on Some Classes of Riemannian and Lorentzian Symmetric Spaces

We compute the Green's function for the Hodge Laplacian on the symmetric spaces M\times\Sigma, where M is a simply connected n-dimensional Riemannian or Lorentzian manifold of constant curvature and \Sigma is a simply connected Riemannian surface of constant curvature. Our approach is based on a generalization to the case of differential forms of the method of spherical means and on the use of Riesz distributions on manifolds. The radial part of the Green's function is governed by a fourth order analogue of the Heun equation.

Classification of marginally trapped surfaces of constant curvature in Lorentzian complex plane

A surface in the Lorentzian complex plane ${\bf C}^2_1$ is called {\it marginally trapped\/} if its mean curvature vector is light-like at each point on the surface. In this article, we classify marginally trapped surfaces of constant curvature in the Lorentzian complex plane ${\bf C}^2_1$. Our main results state that there exist twenty-one families of marginally trapped surfaces of constant curvature in ${\bf C}^2_1$. Conversely, up to rigid motions and dilations, marginally trapped surfaces of constant curvature in ${\bf C}^2_1$ are locally obtained from these twenty-one families.

Jacobi osculating rank and isotropic geodesics on naturally reductive 3-manifolds

We study the Jacobi osculating rank of geodesics on naturally reductive homogeneous manifolds and we apply this theory to the 3-dimensional case. Here, each non-symmetric, simply connected naturally reductive 3-manifold can be given as a principal bundle M 3 ( κ , τ ) over a surface of constant curvature κ, such that the curvature of its horizontal distribution is a constant τ > 0 , with τ 2 ≠ κ . Then, we prove that the Jacobi osculating rank of every geodesic of M 3 ( κ , τ ) is two except for the Hopf fibers, where it is zero. Moreover, we determine all isotropic geodesics and the isotropic tangent conjugate locus.

Hamiltonian-stationary Lagrangian surfaces of constant curvature [formula omitted] in complex space form [formula omitted

A Lagrangian surface in a Kaehler manifold is called Hamiltonian-stationary if it is a critical point of the area functional restricted to compactly supported Hamiltonian variations. In the first part of this article, we study the traveling wave solutions of an over-determined PDE system arising from a study of Hamiltonian-stationary Lagrangian surfaces of constant curvature ε in a complex space form M ̃ 2 ( 4 ε ) . Then, by applying the traveling wave solutions, we construct families of type II Hamiltonian-stationary Lagrangian surfaces of constant curvature ε in complex space forms M ̃ 2 ( 4 ε ) via an effective method of [B.Y. Chen, F. Dillen, L. Verstraelen and L. Vrancken, Lagrangian isometric immersions of a real-space-form M n ( c ) into a complex-space-form M ̃ n ( 4 c ) , Math. Proc. Cambridge Philo. Soc. 124 (1998) 107–125]. In the second part, we are able to completely solve the over-determined PDE system for the case ε = 0 and to determine their corresponding Hamiltonian-stationary Lagrangian surfaces. As an immediate by-product, some interesting new families of Hamiltonian-stationary Lagrangian surfaces of constant curvature are obtained.

The Dynamics of Pendulums on Surfaces of Constant Curvature

We define the notion of a pendulum on a surface of constant curvature and study the motion of a mass at a fixed distance from a pivot. We consider some special cases: first a pivot that moves with constant speed along a geodesic, and then a pivot that undergoes acceleration along a fixed geodesic.

Surfaces of constant curvature in 3-dimensional space forms

Surfaces of constant curvature in 3-dimensional space forms

On the integrability of the generalized Fisher-type nonlinear diffusion equations

In this paper, the geometric integrability and Lax integrability of the generalized Fisher-type nonlinear diffusion equations with modified diffusion in (1+1) and (2+1) dimensions are studied by the pseudo-spherical surface geometry method and prolongation technique. It is shown that the (1+1)-dimensional Fisher-type nonlinear diffusion equation is geometrically integrable in the sense of describing a pseudo-spherical surface of constant curvature −1 only for m = 2, and the generalized Fisher-type nonlinear diffusion equations in (1+1) and (2+1) dimensions are Lax integrable only for m = 2. This paper extends the results in Bindu et al 2001 (J. Phys. A: Math. Gen. 34 L689) and further provides the integrability information of (1+1)- and (2+1)-dimensional Fisher-type nonlinear diffusion equations for m = 2.

Constant Scalar Curvature Kähler Surfaces and Parabolic Polystability

A complex ruled surface admits an iterated blow-up encoded by a parabolic structure with rational weights. Under a condition of parabolic stability, one can construct a Kähler metric of constant scalar curvature on the blow-up according to Rollin and Singer (J. Eur. Math. Soc., 2004). We present a generalization of this construction to the case of parabolically polystable ruled surfaces. Thus, we can produce numerous examples of Kähler surfaces of constant scalar curvature with circle or toric symmetry.

Discrete analog of the projective equivalence and integrable billiard tables

AbstractA class of discrete dynamical systems called projectively (or geodesically) equivalent Lagrangian systems is defined. We prove that these systems admit families of integrals. In the case of geodesically equivalent billiard tables, these integrals are pairwise commuting. We describe a family of geodesically equivalent billiard tables on surfaces of constant curvature. This is a special case of the so-called ‘Liouville billiard tables’.

Lagrangian minimal surfaces in Lorentzian complex plane

Flat Lagrangian minimal surfaces in the Lorentzian complex plane \({\mathbb{C}}^{2}_{1}\) are classified by B. Y. Chen and L. Vrancken in [8]. On the other hand, Vrancken proves in [11] that Lagrangian minimal surfaces of constant curvature in \({\mathbb{C}}^{2}_{1}\) are flat surfaces. In this article, we classify all Lagrangian minimal surfaces in \({\mathbb{C}}^{2}_{1}\) which are free from flat points.

Minimal Lagrangian diffeomorphisms between domains in the hyperbolic plane

Let N be a complete, simply-connected surface of constant curvature \kappa \leq 0. Moreover, suppose that \Omega and \tilde{\Omega} are strictly convex domains in N with the same area. We show that there exists an area-preserving diffeomorphism from \Omega to \tilde{\Omega} whose graph is a minimal submanifold of N \times N.