Euclidean Three-space Research Articles

A Relationship Between Spin and Geometry

In physics, spin is often seen exclusively through the lens of its phenomenological character: as an intrinsic form of angular momentum. However, there is mounting evidence that spin fundamentally originates as a quality of geometry, not of dynamics, and recent work further suggests that the structure of non-relativistic Euclidean three-space is sufficient to define it. In this paper, we directly explicate this fundamentally non-relativistic, geometric nature of spin by constructing non-commutative algebras of position operators which subsume the structure of an arbitrary spin system. These “Spin-s Position Algebras” are defined by elementary means and from the properties of Euclidean three-space alone, and constitute a fundamentally new model for quantum mechanical systems with non-zero spin, within which neither position and spin degrees of freedom, nor position degrees of freedom within themselves, commute. This reveals that the observables of a system with spin can be described completely geometrically as tensors of oriented planar elements, and that the presence of non-zero spin in a system naturally generates a non-commutative geometry within it. We will also discuss the potential for the Spin-s Position Algebras to form the foundation for a generalisation to arbitrary spin of the Clifford and Duffin–Kemmer–Petiau algebras.

Helicoidal minimal surfaces in the 3-sphere: an approach via spherical curves

We prove an existence and uniqueness theorem about spherical helicoidal (in particular, rotational) surfaces with prescribed mean or Gaussian curvature in terms of a continuous function depending on the distance to its axis. As an application in the case of vanishing mean curvature, it is shown that the well-known conjugation between the belicoid and the catenoid in Euclidean three-space extends naturally to the 3-sphere to their spherical versions and determine in a quite explicit way their associated surfaces in the sense of Lawson. As a key tool, we use the notion of spherical angular momentum of the spherical curves that play the role of profile curves of the minimal helicoidal surfaces in the 3-sphere.

The world in a grain of sand: Condensing the string vacuum degeneracy

The proliferation of vacuum solutions to string theory highly deter the search for the Standard Model. We propose a novel approach to this problem by finding an efficient measure of similarity of vacua. Using one million concrete examples, the paradigm of few-shot machine-learning represents them as points in Euclidean three-space with similar points clustered together. We thereby compress the search space for desired physics to within one percent of the original. Our analysis provides an explicit method for machine-learning the landscape and finding ‘typicality’, even with minuscule data.

A Surface Pencil with Bertrand Curves as Joint Curvature Lines in Euclidean Three-Space

The main outcome of this work is the construction of a surface pencil with a similarity to Bertrand curves in Euclidean 3-space E3. Then, by exploiting the Serret–Frenet frame, we deduce the sufficient and necessary conditions for a surface pencil with Bertrand curves as joint curvature lines. Consequently, the expansion to the ruled surface pencil is also designed. As demonstrations of our essential findings, we illustrate some models to emphasize the process.

A new proof of Atiyah’s conjecture on configurations of four points

In the article entitled Atiyah, “The geometry of classical particles,” Surv. Differ. Geom. 7, 1–15 (2000) and in the article entitled Atiyah, “Configurations of points,” Philos. Trans. R. Soc., A 359, 1375–1387 (2001), Sir Michael Atiyah introduced what is known as the Atiyah problem on configurations of points, which can be briefly described as the conjecture that the n polynomials (each defined up to a phase factor) associated geometrically to a configuration of n distinct points in R3 are always linearly independent. The first “hard” case is for n = 4 points, for which the linear independence conjecture was proved by Eastwood and Norbury- in the article entitled “A proof of Atiyah’s conjecture on configurations of four points in Euclidean three-space,” Geom. Topol. 5(2), 885–893 (2001). We present a new proof of Atiyah’s linear independence conjecture on configurations of four points, i.e., of Eastwood and Norbury’s theorem. Our proof consists in showing that the Gram matrix of the four polynomials associated with a configuration of four points in Euclidean 3-space is always positive definite. It makes use of 2-spinor calculus and the theory of Hermitian positive semidefinite matrices.

Clairaut surfaces in Euclidean three-space

In this note, we investigate local isometric immersion of Clairaut metrics in Euclidean three-space. A Clairaut metric is determined up to isometry by a single function of one variable. We show that an isometric immersion is formally determined by two functions of one variable, uniquely up to coordinate reflection and ambient Euclidean motions. Further, if the Clairaut metric and these two functions are real-analytic, there exists a local isometric immersion realizing these data. We give a more explicit description for a finite-dimensional family of examples.

Rigidity of minimal Lagrangian diffeomorphisms between spherical cone surfaces

We prove that any minimal Lagrangian diffeomorphism between two closed spherical surfaces with cone singularities is an isometry, without any assumption on the multiangles of the two surfaces. As an application, we show that every branched immersion of a closed surface of constant positive Gaussian curvature in Euclidean three-space is a branched covering onto a round sphere, thus generalizing the classical rigidity theorem of Liebmann to branched immersions.

Distribution Characteristics of Screw Axes Belonging to General Three-System of Screws

The cylindroid describes the exact distribution of the ∞1 screw axes of the general two-system of screws. For the general three-system, this article introduces the surface on which lies the position vector endpoints of its ∞2 screw axes, and then develops the finite distribution zone in which they most densely distribute. In particular, this paper explores the inner structure of the zone by partitioning the general three-system into ∞1 two-subsystems, and then reveals the principle of constructing the distribution space of the general three-system by that of the general two-system in Euclidean three-space. In pursuit of a generalized decomposition method, we propose the varying-pitch ruled surface carrying the screws whose pitch value continuously varies according to any desired function rule, based on the pitch-hyperboloid carrying all the equal-pitch screws. This investigation provides some useful guides for the axode planning and then the motion planning of three degree-of-freedom parallel mechanisms.

Bounds for the Morse index of free boundary minimal surfaces

Inspired by work of Ejiri-Micallef on closed minimal surfaces, we compare the energy index and the area index of a free-boundary minimal surface of a Riemannian manifold with boundary, and show that the area index is controlled from above by the area and the topology of the surface. Combining these results with work of Fraser-Li, we conclude that the area index of a free-boundary minimal surface in a convex domain of Euclidean three-space, is bounded from above by a linear function of its genus and its number of boundary components. We also prove index bounds for submanifolds of higher dimension.

Geometric inequalities involving mean curvature for closed surfaces

In this paper we prove some geometric inequalities for closed surfaces in Euclidean three-space. Motivated by Gage's inequality for convex curves, we first verify that for convex surfaces the Willmore energy is bounded below by some scale-invariant quantities. In particular, we obtain an optimal scaling law between the Willmore energy and the isoperimetric ratio under convexity. In addition, we also address Topping's conjecture relating diameter and mean curvature for connected closed surfaces. We prove this conjecture in the class of simply-connected axisymmetric surfaces, and moreover obtain a sharp remainder term which ensures the first evidence that optimal shapes are necessarily straight even without convexity.

Special Liouville metric with the Ricci condition

Two necessary conditions for the induced metrics of parallel mean curvature surfaces in a complex space form of complex two-dimension are observed. One is similar to the Ricci condition of the classical surface theory in Euclidean three-space and the other is related to the Liouville metric. Conversely, we prove that a special type of the Liouville metric on a domain in the Euclidean two-plane whose Gaussian curvature satisfies the differential equation similar to the Ricci condition is explicitly determined by an elliptic function. We have isometric immersions from a simply connected two-dimensional Riemannian manifold with the special type of the Liouville metric satisfying the Ricci condition to the complex hyperbolic plane with parallel mean curvature vector.

Schwarzite nets: a wealth of 3-valent examples sharing similar topologies and symmetries

We enumerate trivalent reticulations of two- and three-periodic hyperbolic surfaces by equal-sided n -gonal faces, ( n , 3), where n = 7, 8, 9, 10, 12. These are the simplest hyperbolic generalizations of the planar graphene net, (6, 3) and dodecahedrane, (5, 3). The enumeration proceeds by deleting isometries of regular reticulations of two-dimensional hyperbolic space until the ( n , 3) nets can be embedded in euclidean three-space via periodic hyperbolic surfaces. Those nets are then symmetrized in euclidean space retaining their net topology, leading to explicit crystallographic net embeddings whose edges are as equal as possible, affording candidtae patterns for graphitic schwarzites. The resulting schwarzites are the most symmetric examples. More than one hundred topologically distinct nets are described, most of which are novel.

On congruences of Laguerre cycles and their envelopes

In the cyclographic model of Laguerre geometry, the isotropy projection establishes a correspondence between submanifolds in the Lorentz (n + 1)-space and oriented hypersphere congruences in the Euclidean n-space. In this paper, we investigate the relation between the Lorentzian geometry of the sphere congruences and the Euclidean geometry of the corresponding oriented envelopes. We introduce the notion and discuss the existence of marginally outer trapped submanifolds associated to an (n − 1)-dimensional Legendre submanifold of the unit tangent bundle over the Euclidean n-space. This extends the notion of Laguerre–Gauss map of an oriented surface. Two special classes of marginally outer trapped surfaces are characterized in terms of their oriented envelopes. The Lorentzian geometry of several families of spheres whose envelopes are well-known surfaces in the Euclidean three-space is investigated.

Surfaces of Revolution and Canal Surfaces with Generalized Cheng–Yau 1-Type Gauss Maps

In the present work, the notion of generalized Cheng–Yau 1-type Gauss map is proposed, which is similar to the idea of generalized 1-type Gauss maps. Based on this concept, the surfaces of revolution and the canal surfaces in the Euclidean three-space are classified. First of all, we show that the Gauss map of any surfaces of revolution with a unit speed profile curve is of generalized Cheng–Yau 1-type. At the same time, an oriented canal surface has a generalized Cheng–Yau 1-type Gauss map if, and only if, it is an open part of a surface of revolution or a torus.

GAUSSIAN CURVATURE AND UNICITY PROBLEM OF GAUSS MAPS OF VARIOUS CLASSES OF SURFACES

In this article, we establish a new estimate for the Gaussian curvature of open Riemann surfaces in Euclidean three-space with a specified conformal metric regarding the uniqueness of the holomorphic maps of these surfaces. As its applications, we give new proofs on the unicity problems for the Gauss maps of various classes of surfaces, in particular, minimal surfaces in Euclidean three-space, constant mean curvature one surfaces in the hyperbolic three-space, maximal surfaces in the Lorentz–Minkowski three-space, improper affine spheres in the affine three-space and flat surfaces in the hyperbolic three-space.

Spectral Flow for Dirac Operators with Magnetic Links

This paper is devoted to the study of the spectral properties of Dirac operators on the three-sphere with singular magnetic fields supported on smooth, oriented links. As for Aharonov–Bohm solenoids in the Euclidean three-space, the flux carried by an oriented knot features a $$2\pi $$-periodicity of the associated operator. For a given link, one thus obtains a family of Dirac operators indexed by a torus of fluxes. We study the spectral flow of paths of such operators corresponding to loops in this torus. The spectral flow is in general nontrivial. In the special case of a link of unknots, we derive an explicit formula for the spectral flow of any loop on the torus of fluxes. It is given in terms of the linking numbers of the knots and their writhes.

QA-deformation of surface of negative Gaussian curvature

An infinitesimal deformation with the given law of changing the element of area of a surface in Euclidean three-space was considered in this article. Such deformation in the article was called the quasiareal infinitesimal deformation or, briefly, the QA-deformation. The problem of finding the QA-deformation, under which the unit normal vector to the surface is preserved, was reduced to the study of one nonhomogeneous partial differential equation of the second order with respect to one unknown function. The initial conditions, under which the only one QA-deformation with the stationary unit normal vector exists, were defined for the surfaces of a negative Gaussian curvature. In this case, for the above equation, the Cauchy and Goursat problems were applied. The initial conditions of these tasks were expressed through the deforming vector.

Semi-Discrete Constant Mean Curvature Surfaces of Revolution in Minkowski Space

In this paper we describe semi-discrete isothermic constant mean curvature surfaces of revolution with smooth profile curves in Minkowski three-space. Unlike the case of semi-discrete constant mean curvature surfaces in Euclidean three-space, they might have certain types of singularities in a sense defined by the second author in a previous work. We analyze the singularities of such surfaces.

An explicit reconstruction algorithm for the transverse ray transform of a second rank tensor field from three axis data

We give an explicit plane-by-plane filtered back-projection reconstruction algorithm for the transverse ray transform of symmetric second rank tensor fields on Euclidean three-space, using data from rotation about three orthogonal axes. We show that in the general case two-axis data is insufficient, but we give an explicit reconstruction procedure for the potential case with two-axis data. We describe a numerical implementation of the three-axis algorithm and give reconstruction results for simulated data.

On the local isometric embedding in ℝ3 of surfaces with zero sets of Gaussian curvature forming cusp domains

ABSTRACTWe study the problem of isometrically embedding a two-dimensional Riemannian manifold into Euclidean three-space. It is shown that if the Gaussian curvature vanishes to finite order and its zero set consists of two smooth curves tangent at a point, then local sufficiently smooth isometric embeddings exist.