Euclidean Three-space Research Articles

Function-theoretic Properties for the Gauss Maps of Various Classes of Surfaces

AbstractWe elucidate the geometric background of function-theoretic properties for the Gauss maps of several classes of immersed surfaces in three-dimensional space forms, for example, minimal surfaces in Euclidean three-space, improper affine spheres in the affine three-space, and constant mean curvature one surfaces and flat surfaces in hyperbolic three-space. To achieve this purpose, we prove an optimal curvature bound for a specified conformal metric on an open Riemann surface and give some applications. We also provide unicity theorems for the Gauss maps of these classes of surfaces.

Ropelength criticality

The ropelength problem asks for the minimum-length configuration of a knotted diameter-one tube embedded in Euclidean three-space. The core curve of such a tube is called a tight knot, and its length is a knot invariant measuring complexity. In terms of the core curve, the thickness constraint has two parts: an upper bound on curvature and a self-contact condition. We give a set of necessary and sufficient conditions for criticality with respect to this constraint, based on a version of the Kuhn-Tucker theorem that we established in previous work. The key technical difficulty is to compute the derivative of thickness under a smooth perturbation. This is accomplished by writing thickness as the minimum of a $C^1$-compact family of smooth functions in order to apply a theorem of Clarke. We give a number of applications, including a classification of the "supercoiled helices" formed by critical curves with no self-contacts (constrained by curvature alone) and an explicit but surprisingly complicated description of the "clasp" junctions formed when one rope is pulled tight over another.

SMALL PERTURBATIONS AND INFINITESIMAL DEFORMATIONS ON SURFACES OF REVOLUTION

Any given surface of revolution embedded in Euclidean three-space can always be perturbed by arbitrarily small ambient isotopies as to admit highly nontrivial vector fields inducing infinitesimal deformations. For this matter Morse Theory is used, clarifying and giving a generalization of a problem originaly intro- duced by M. Spivak (10) in a modern perspective.

Polar Decomposition of Unit Dual Quaternions

This paper seeks to extend the notion of polar decomposition (PD) from matrix algebra to dual-quaternion algebra. The goal is to obtain a simple, efficient and explicit method for determining the PD of spatial displacements in Euclidean three-space that belong to a special Euclidean Group known as SE(3). It has been known that such a decomposition is equivalent to the projection of an element of SE(3) onto SO(4) that yields hyper spherical displacements that best approximate rigid-body displacements. It is shown in this paper that a dual quaternion representing an element of SE(3) can be decomposed into a pair of unit quaternions, called double quaternion, that represents an element of SO(4). Furthermore, this decomposition process may be interpreted as the projection of a point in four-dimensional space onto a unit hypersphere. An example is provided to illustrate that the results obtained from this dual-quaternion based polar decomposition are same as those obtained from the matrix based polar decomposition.

On the maximal number of exceptional values of Gauss maps for various classes of surfaces

The main goal of this paper is to reveal the geometric meaning of the maximal number of exceptional values of Gauss maps for several classes of immersed surfaces in space forms, for example, complete minimal surfaces in the Euclidean three-space, weakly complete improper affine spheres in the affine three-space and weakly complete flat surfaces in the hyperbolic three-space. For this purpose, we give an effective curvature bound for a specified conformal metric on an open Riemann surface.

Clifford Algebras in Symplectic Geometry and Quantum Mechanics

The necessary appearance of Clifford algebras in the quantum description of fermions has prompted us to re-examine the fundamental role played by the quaternion Clifford algebra, C 0,2. This algebra is essentially the geometric algebra describing the rotational properties of space. Hidden within this algebra are symplectic structures with Heisenberg algebras at their core. This algebra also enables us to define a Poisson algebra of all homogeneous quadratic polynomials on a two-dimensional sub-space, \(\mathbb{F}^{a}\) of the Euclidean three-space. This enables us to construct a Poisson Clifford algebra, ℍ F , of a finite dimensional phase space which will carry the dynamics. The quantum dynamics appears as a realisation of ℍ F in terms of a Clifford algebra consisting of Hermitian operators.

Rigidity for Multi-Taub-NUT metrics

This paper provides a classification result for gravitational instantons with cubic volume growth and cyclic fundamental group at infinity. It proves that a complete hyperkahler manifold asymptotic to a circle fibration over the Euclidean three-space is either the standard R 3 ◊ S 1 or a Multi-Taub-NUT manifold. In par- ticular, the underlying complex manifold is either C ◊ C/Z or a minimal resolution of a cyclic Kleinian singularity.

Topological uniqueness of negatively curved surfaces

AbstractIn this paper we consider complete, noncompact, negatively curved surfaces that are twice continuously differentiably embedded in Euclidean three-space, showing that if such surfaces have square integrable second fundamental form, then their topology must, by the index method, be an annulus. We then show how this relates to some minimal surface theorems and has a corollary on minimal surfaces with finite total curvature. In addition, we discuss, by the index method, the relation between the topology and asymptotic curves. Finally, we apply the results yielded to the problem of isometrical immersions into Euclidean three-space of black hole models.

Topological uniqueness of negatively curved surfaces

AbstractIn this paper we consider complete, noncompact, negatively curved surfaces that are twice continuously differentiably embedded in Euclidean three-space, showing that if such surfaces have square integrable second fundamental form, then their topology must, by the index method, be an annulus. We then show how this relates to some minimal surface theorems and has a corollary on minimal surfaces with finite total curvature. In addition, we discuss, by the index method, the relation between the topology and asymptotic curves. Finally, we apply the results yielded to the problem of isometrical immersions into Euclidean three-space of black hole models.

On the asymptotic geometry of gravitational instantons

We investigate the geometry at infinity of the so-called instan- tons, i.e. asymptotically flat hyperkahler four-manifolds, in relation with their volume growth. In particular, we prove that gravitational instantons with cubic volume growth are ALF, namely asymptotic to a circle fibration over a Euclidean three-space, with fibers of asymptotically constant length.

Three-vector and scalar field identities and uniqueness theorems in Euclidean and Minkowski spaces

Euclidean three-space and Minkowski four-space identities and uniqueness theorems are reviewed and extended. A Helmholtz identity is used to prove two three-vector uniqueness theorems in Euclidean three-space. The first theorem specifies the divergence and curl of the vector, and the second is a Helmholtz type theorem that sums the irrotational and solenoidal parts of the vector. The second theorem is shown to be valid for three-vector fields that are time dependent. A time-dependent extension of the Helmholtz identity is also derived. However, only the three-vector and scalar components of a Minkowski space four-vector identity are shown to yield two identities that lead to a uniqueness theorem of the first or source type. Also, the field equations of this latter theorem appear to be sufficiently general such that the field equations naturally divide into two distinct classes, a four-solenoidal electromagnetic type class in a relativistic transverse gauge and a four-irrotational class in a relativistic longitudinal gauge.

Isometric Embeddings into the Minkowski Space and New Quasi-Local Mass

The definition of quasi-local mass for a bounded space-like region in space-time is essential in several major unsettled problems in general relativity. The quasi-local mass is expected to be a type of flux integral on the boundary two-surface and should be independent of whichever space-like region it bounds. An important idea which is related to the Hamiltonian formulation of general relativity is to consider a reference surface in a flat ambient space with the same first fundamental form and derive the quasi-local mass from the difference of the extrinsic geometries. This approach has been taken by Brown-York and Liu-Yau (see also related works) to define such notions using the isometric embedding theorem into the Euclidean three-space. However, there exist surfaces in the Minkowski space whose quasilocal mass is strictly positive. It appears that the momentum information needs to be accounted for to reconcile the difference. In order to fully capture this information, we use isometric embeddings into the Minkowski space as references. In this article, we first prove an existence and uniqueness theorem for such isometric embeddings. We then solve the boundary value problem for Jang's equation as a procedure to recognize such a surface in the Minkowski space. In doing so, we discover new expression of quasi-local mass. The new mass is positive when the ambient space-time satisfies the dominant energy condition and vanishes on surfaces in the Minkowski space. It also has the nice asymptotic behavior at spatial and null infinity. Some of these results were announced in [29].

Dynamische Systeme

This workshop, organised by Hakan Eliasson (Paris), Helmut Hofer (New York), and Jean-Christophe Yoccoz (Paris) continued the biannual series at Oberwolfach on Dynamical Systems that started as the “Moser & Zehnder meeting” in 1981. The workshop was attended by more than 50 participants from 12 countries. The main themes of the workshop were the new results and developments in the area of classical dynamical systems, in particular in celestial mechanics and Hamiltonian systems. The workshop covers a large area of dynamical systems and the following samples give an idea about the scope. The topic of Arnold Diffusion was treated in great detail by talks of M. Levi and J. Mather. In the classical field of celestial mechanics new insight has been gained about two interesting families of relative periodic solutions of the spatial n-body problem (the P_{12} -family and the hip-hop-family) as they share the property of being global continuations of Lyapunov families which bifurcate from a relative equilibrium solution in the direction orthogonal to the plane of motion (A. Chenciner). K. Kuperberg reported on the construction of flows on three-manifolds where every nonconstant trajectory is wild in a sense related to the Artin–Fox example of an exotic arc in Euclidean three-space. Other results were concerned with one of the main problems in Hamiltonian dynamic, namely the stability of motions in nearly-integrable systems (L. Niedermann). Y. Pesin outlined the construction of hyperbolic volume-preserving flows on manifolds of dimension at least three and V. Ginzburg described the recent developments concerning the Conley Conjecture for periodic points of Hamiltonian symplectic maps. John Franks described his recent results about group actions on surfaces. In addition several talks covered new Floer-theoretic methods in the study of Hamiltonian systems and it will be interesting to see in the future how these symplectic methods can be merged with the more classical dynamical systems methods.

Hamiltonian structure for a neutrally buoyant rigid body interacting with N vortex rings of arbitrary shape: the case of arbitrary smooth body shape

We present a (noncanonical) Hamiltonian model for the interaction of a neutrally buoyant, arbitrarily shaped smooth rigid body with N thin closed vortex filaments of arbitrary shape in an infinite ideal fluid in Euclidean three-space. The rings are modeled without cores and, as geometrical objects, viewed as N smooth closed curves in space. The velocity field associated with each ring in the absence of the body is given by the Biot–Savart law with the infinite self-induced velocity assumed to be regularized in some appropriate way. In the presence of the moving rigid body, the velocity field of each ring is modified by the addition of potential fields associated with the image vorticity and with the irrotational flow induced by the motion of the body. The equations of motion for this dynamically coupled body-rings model are obtained using conservation of linear and angular momenta. These equations are shown to possess a Hamiltonian structure when written on an appropriately defined Poisson product manifold equipped with a Poisson bracket which is the sum of the Lie–Poisson bracket from rigid body mechanics and the canonical bracket on the phase space of the vortex filaments. The Hamiltonian function is the total kinetic energy of the system with the self-induced kinetic energy regularized. The Hamiltonian structure is independent of the shape of the body, (and hence) the explicit form of the image field, and the method of regularization, provided the self-induced velocity and kinetic energy are regularized in way that satisfies certain reasonable consistency conditions.

On Rectifying Dual Space Curves

We give some characterizations of the rectifying curves in the dual space and show that rectifying dual space curves can be stated with the aid of dual unit spherical curves. Thus, we have a link between rectifying dual space curves and classical surfaces in the Euclidean three-space.

Generalized surfaces with constant H/K in Euclidean three-space

Surfaces in Euclidean three-space with constant ratio of mean curvature to Gauss curvature arise naturally as the parallel surfaces to minimal surfaces. They might possess singularities which occur naturally as focal points of minimal surfaces. We study geometric properties and the singularities of such surfaces, prove some global results about them, and provide a Bjorling formula to construct such surfaces with prescribed point or curve singularities.

Erratum: “A class of superintegrable systems of Calogero type” [J. Math. Phys. 47, 093505 (2006)

We show that the three body Calogero model with inverse square potentials can be interpreted as a maximally superintegrable and multiseparable system in Euclidean three-space. As such it is a special case of a family of systems involving one arbitrary function of one variable.

On the Fixed Homogeneous Circle Problem

Abstract We give some results about the dynamics of a particle moving in Euclidean three-space under the influence of the gravitational force induced by a fixed homogeneous circle. Our main results concern (1) singularities and (2) the dynamics in the plane that contains the circle. The study presented here is purely analytic.

On the existence of periodic orbits for the fixed homogeneous circle problem

We prove the existence of some types of periodic orbits for a particle moving in Euclidean three-space under the influence of the gravitational force induced by a fixed homogeneous circle. These types include periodic orbits very far and very near the homogeneous circle, as well as eight and spiral periodic orbits.

A class of superintegrable systems of Calogero type

We show that the three-body Calogero model with inverse square potentials can be interpreted as a maximally superintegrable and multiseparable system in Euclidean three-space. As such it is a special case of a family of systems involving one arbitrary function of one variable.