Non-compact Surfaces Research Articles

Existence of solution of the logarithmic diffusion equation with bounded above Gauss curvature

We give a simple proof of an extension of the existence results of Ricci flow of Giesen and Topping (2010, 2011) [15,20], on incomplete surfaces with bounded above Gauss curvature without using the difficult Shi’s existence theorem of Ricci flow on complete non-compact surfaces and the pseudolocality theorem of Perelman [7] on Ricci flow. We will also give a simple proof of a special case of the existence theorem of Topping (2010) [16] without using the existence theorem of Shi (1989) [9].

Ricci flows with unbounded curvature

We show that any noncompact Riemann surface admits a complete Ricci flow g(t), t\in[0,\infty), which has unbounded curvature for all t\in[0,\infty).

Short Loop Decompositions of Surfaces and the Geometry of Jacobians

Given a Riemannian surface, we consider a naturally embedded graph which captures part of the topology and geometry of the surface. By studying this graph, we obtain results in three different directions. First, we find bounds on the lengths of homologically independent curves on closed Riemannian surfaces. As a consequence, we show that for any $${\lambda \in (0, 1)}$$ there exists a constant C λ such that every closed Riemannian surface of genus g whose area is normalized at 4π (g – 1) has at least [λg] homologically independent loops of length at most C λ log(g). This result extends Gromov’s asymptotic log(g) bound on the homological systole of genus g surfaces. We construct hyperbolic surfaces showing that our general result is sharp. We also extend the upper bound obtained by P. Buser and P. Sarnak on the minimal norm of nonzero period lattice vectors of Riemann surfaces in their geometric approach of the Schottky problem to almost g homologically independent vectors. Then, we consider the lengths of pants decompositions on complete Riemannian surfaces in connexion with Bers’ constant and its generalizations. In particular, we show that a complete noncompact Riemannian surface of genus g with n ends and area normalized to $${4\pi(g + \frac{n}{2}-1)}$$ admits a pants decomposition whose total length (sum of the lengths) does not exceed C g n log(n + 1) for some constant C g depending only on the genus. Finally, we obtain a lower bound on the systolic area of finitely presentable nontrivial groups with no free factor isomorphic to $${\mathbb{Z}}$$ in terms of its first Betti number. The asymptotic behavior of this lower bound is optimal.

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.

Eigenvalue estimates for hypersurfaces in ℍm× ℝ and applications

the operator LD1gC aC bK g on a complete noncompact surface .M 2 ; g/. Assuming that L is nonnegative for some constants a > 0 and b > 1=4, we show that the infimum of the spectrum of M 2 is bounded from above by a=.4b 1/. We apply this result to stable minimal surfaces immersed into homogeneous 3-manifolds.

Ideal tilings and symbolic dynamics for negatively curved surfaces

AbstractLet X be a non-compact surface (or 2-orbifold) of finite type with a metric of strictly negative curvature. Using an ideal tiling of the universal cover of X, we give a symbolic description of the recurrent geodesics on X. This extends Series’s method of coding geodesics on the modular surface.

Capillary Floating and the Billiard Ball Problem

We study the recurrence and ergodicity for the billiard on noncompact polygonal surfaces with a free, cocompact action of $\Z$ or $\Z^2$. In the $\Z$-periodic case, we establish criteria for recurrence. In the more difficult $\Z^2$-periodic case, we establish some general results. For a particular family of $\Z^2$-periodic polygonal surfaces, known in the physics literature as the wind-tree model, assuming certain restrictions of geometric nature, we obtain the ergodic decomposition of directional billiard dynamics for a dense, countable set of directions. This is a consequence of our results on the ergodicity of $\ZZ$-valued cocycles over irrational rotations.

Limit Cycles Bifurcating from a 2-Dimensional Isochronous Torus in ℝ3

Abstract In this paper we illustrate the explicit implementation of a method for computing limit cycles which bifurcate from a 2-dimensional isochronous set contained in ℝ3, when we perturb it inside a class of differential systems. This method is based in the averaging theory. As far as we know all applications of this method have been made perturbing noncompact surfaces, as for instance a plane or a cylinder in ℝ3. Here we consider polynomial perturbations of degree d of an isochronous torus. We prove that, up to first order in the perturbation, at most 2(d+1) limit cycles can bifurcate from a such torus and that there exist polynomial perturbations of degree d of the torus such that exactly ν limit cycles bifurcate from such a torus for every ν ∈ {2, 4,...,2(d + 1)}.

Eigenvalues of Laplacian with Constant Magnetic Field on Non-Compact Hyperbolic Surfaces with Finite Area

We consider a magnetic Laplacian $-\Delta_A=(id+A)^\star (id+A)$ on a noncompact hyperbolic surface $\mM $ with finite area. $A$ is a real one-form and the magnetic field $dA$ is constant in each cusp. When the harmonic component of $A$ satifies some quantified condition, the spectrum of $-\Delta_A$ is discrete. In this case we prove that the counting function of the eigenvalues of $-\Delta_{A}$ satisfies the classical Weyl formula, even when $dA=0. $

Maps related to Grigorchuk’s group

Grigorchuk’s group of intermediate growth can be represented, through its action on the infinite binary rooted tree, as the automorphism group of a regular map G on a non-compact surface. A theory of growth of maps is developed, and it is shown that G has intermediate growth. Some compact and non-compact quotients of G are described, and it is shown how these ideas may be extended to the generalised Grigorchuk groups.

Total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below: I

We investigate the finiteness structure of a complete non-compact $n$-dimensional Riemannian manifold $M$ whose radial curvature at a base point of $M$ is bounded from below by that of a non-compact von Mangoldt surface of revolution with its total curvature greater than $\pi$. We show, as our main theorem, that all Busemann functions on $M$ are exhaustions, and that there exists a compact subset of $M$ such that the compact set contains all critical points for any Busemann function on $M$. As corollaries by the main theorem, $M$ has finite topological type, and the isometry group of $M$ is compact.

Geometry, topology and dynamics of geodesic flows on noncompact polygonal surfaces

We establish the background for the study of geodesics on noncompact polygonal surfaces. For illustration, we study the recurrence of geodesics on $Z$-periodic polygonal surfaces. We prove, in particular, that almost all geodesics on a topologically typical $Z$-periodic surface with boundary are recurrent.

Total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below. II

We prove, as our main theorem, the finiteness of topological type of a complete open Riemannian manifold $M$ with a base point $p \in M$ whose radial curvature at $p$ is bounded from below by that of a non-compact model surface of revolution $\widetilde {M}$ which admits a finite total curvature and has no pair of cut points in a sector. Here a sector is, by definition, a domain cut off by two meridians emanating from the base point $\tilde {p} \in \widetilde {M}$. Notice that our model $\widetilde {M}$ does not always satisfy the diameter growth condition introduced by Abresch and Gromoll. In order to prove the main theorem, we need a new type of the Toponogov comparison theorem. As an application of the main theorem, we present a partial answer to Milnor’s open conjecture on the fundamental group of complete open manifolds.

Affine deformations of a three-holed sphere

Associated to every complete affine 3–manifold M with nonsolvable fundamental group is a noncompact hyperbolic surface Σ. We classify these complete affine structures when Σ is homeomorphic to a three-holed sphere. In particular, for every such complete hyperbolic surface Σ, the deformation space identifies with two opposite octants in ℝ3. Furthermore every M admits a fundamental polyhedron bounded by crooked planes. Therefore M is homeomorphic to an open solid handlebody of genus two. As an explicit application of this theory, we construct proper affine deformations of an arithmetic Fuchsian group inside Sp(4,ℤ).

Geometry, topology and dynamics of geodesic flows on noncompact polygonal surfaces

Geometry, topology and dynamics of geodesic flows on noncompact polygonal surfaces

Symplectic critical surfaces in Kähler surfaces

Let M be a Kähler surface and Σ be a closed symplectic surface which is smoothly immersed in M . Let α be the Kähler angle of Σ in M . We first deduce the Euler–Lagrange equation of the functional L = \int_Σ \frac 1 {\cos α} dµ in the class of symplectic surfaces. It is \cos^3 αH = (J(J∇ \cos α)^⊤)^⊥ , where H is the mean curvature vector of Σ in M , and J is the complex structure compatible with the Kähler form ω in M ; it is an elliptic equation. We call a surface satisfying a this equation a symplectic critical surface. We show that, if M is a Kähler–Einstein surface with a nonnegative scalar curvature, each symplectic critical surface is holomorphic. We also study the topological properties of symplectic critical surfaces. By our formula and Webster’s formula, we deduce that the Kähler angle of a compact symplectic critical surface is constant, which is not true a for noncompact symplectic critical surfaces.

The isoperimetric profile of infinite genus surfaces

It is known that if the isoperimetric profile of a finite genus non-compact surface grows faster than \({\sqrt{t}}\) , then it grows at least as a linear function. In other words there are ‘gaps’ in the isoperimetric profile of surfaces with finite genus. In this paper we show that no gap exists for surfaces of infinite genus.

Slow Plasma Dynamo Driven by Electric Current Helicity in Non-Compact Riemann Surfaces of Negative Curvature

Boozer addressed the role of magnetic helicity in dynamos [1]. He pointed out that the magnetic helicity conservation implies that the dynamo action is more easily attainable if the electric potential varies over the surface of the dynamo. This provided motivated us to investigate dynamos in Riemannian curved surfaces [2]. Thiffeault and Boozer [3] discussed the onset of dissipation in kinematic dynamos. In this paper, when curvature is constant and negative, a simple laminar dynamo solution is obtained on the flow topology of a Poincare disk, whose Gauss curvature is K = –1. By considering a laminar plasma dynamo [4] the electric current helicity λ ≈ 2.34 m–1 for a Reynolds magnetic number of Rm ≈ 210 and a growth rate of magnetic field |γ| ≈ 0.022 are obtained. Negative constant curvature non-compact H2 manifold, has also been used in onecomponent electron 2D plasma by Fantoni and Tellez [5]. Chicone et al. (CMP (1997)) showed fast dynamos can be supported in compact H2. PACS: 47.65.Md.

The Nekrasov Conjecture for Toric Surfaces

The Nekrasov conjecture predicts a relation between the partition function for N = 2 supersymmetric Yang–Mills theory and the Seiberg-Witten prepotential. For instantons on \({\mathbb{R}^4}\), the conjecture was proved, independently and using different methods, by Nekrasov-Okounkov and Nakajima-Yoshioka. We prove a generalized version of the conjecture for instantons on noncompact toric surfaces.

Gravitating opposites attract

Generalizing previous work by two of the authors, we prove the non-existence of certain stationary configurations in general relativity having a spatial reflection symmetry across a non-compact surface disjoint from the matter region. Our results cover cases such as that of two symmetrically arranged rotating bodies with anti-aligned spins in n + 1 (n ⩾ 3) dimensions, or two symmetrically arranged static bodies with opposite charges in (3 + 1) dimensions. They also cover certain symmetric configurations in (3 + 1)-dimensional gravity coupled to a collection of scalars and Abelian vector fields, such as those that arise in supergravity and Kaluza–Klein models. We also treat the bosonic sector of simple supergravity in 4 + 1 dimensions.