Three-dimensional Space Forms Research Articles

A new geometric flow on 3-manifolds: the K-flow

We define a new geometric flow, which we shall call the K-flow, on 3-dimensional Riemannian manifolds; and study the behavior of Thurston’s model geometries under this flow both analytically and numerically. As an example, we show that an initially arbitrarily deformed homogeneous 3-sphere flows into a round 3-sphere and shrinks to a point in the unnormalized flow; or stays as a round 3-sphere in the volume normalized flow. The K-flow equation arises as the gradient flow of a specific purely quadratic action functional that has appeared as the quadratic part of New Massive Gravity in physics; and a decade earlier in the mathematics literature, as a new variational characterization of three-dimensional space forms. We show the short-time existence of the K-flow using a DeTurck-type argument.

Two generalizations of an over-determined system on a surface

We introduce and study two generalizations of an over-determined system on a surface. The first generalization is an over-determined system of polynomial type, which is in the form of [Formula: see text], where each [Formula: see text] is a one-form; the second generalization is represented as [Formula: see text], [Formula: see text], where [Formula: see text], [Formula: see text] are given by [Formula: see text] for a one-form [Formula: see text] and [Formula: see text] is given by [Formula: see text] for a nowhere zero two-form [Formula: see text]. Studying these systems, we will obtain generalizations and analogs of results we already have on over-determined systems on surfaces. In addition, we will obtain new kinds of results on the systems. Over-determined systems of the first generalization appear on admissible hypersurfaces in [Formula: see text] ([Formula: see text]). Results on the second generalization yield corollaries for over-determined systems on surfaces in three-dimensional non-flat space forms.

On polyharmonic helices in space forms

In this article, we study polyharmonic curves of constant curvature where we mostly focus on the case of curves on the sphere. We classify polyharmonic curves of constant curvature in three-dimensional space forms and derive an explicit family of polyharmonic curves on the sphere. Our results give new insights into the geometric structure of higher order variational problems.

Polyharmonic hypersurfaces into pseudo-Riemannian space forms

In this paper, we shall assume that the ambient manifold is a pseudo-Riemannian space form N^{m+1}_t(c) of dimension m+1 and index t (mge 2 and 1 le tle m). We shall study hypersurfaces {M}^{m}_{t'} which are polyharmonic of order r (briefly, r-harmonic), where rge 3 and either t'=t or t'=t-1. Let A denote the shape operator of {M}^{m}_{t'}. Under the assumptions that {M}^{m}_{t'} is CMC and {{rm{Tr}}},{A}^{2} is a constant, we shall obtain the general condition which determines that {M}^{m}_{t'} is r-harmonic. As a first application, we shall deduce the existence of several new families of proper r-harmonic hypersurfaces with diagonalizable shape operator, and we shall also obtain some results in the direction that our examples are the only possible ones provided that certain assumptions on the principal curvatures hold. Next, we focus on the study of isoparametric hypersurfaces whose shape operator is non-diagonalizable and also in this context we shall prove the existence of some new examples of proper r-harmonic hypersurfaces (r ge 3). Finally, we shall obtain the complete classification of proper r-harmonic isoparametric pseudo-Riemannian surfaces into a three-dimensional Lorentz space form.

Free boundary constant mean curvature surfaces in a strictly convex three-manifold

Let C be a strictly convex domain in a three-dimensional Riemannian manifold with sectional curvature bounded above by a constant, and let $$\Sigma $$ be a constant mean curvature surface with free boundary in C. We provide a pinching condition on the length of the traceless second fundamental form on $$\Sigma $$ which guarantees that the surface is homeomorphic to either a disk or an annulus. Furthermore, under the same pinching condition, we prove that if C is a geodesic ball of three-dimensional space forms, then $$\Sigma $$ is either a spherical cap or a Delaunay surface.

On the umbilic set of immersed surfaces in three-dimensional space forms

We prove that, under some integral inequality regarding the mean curvature and the traceless second fundamental form, the set of umbilical points of an immersed surface in a 3-dimensional space form has positive measure. In case of an immersed sphere our result can be seen as a generalization of the celebrated Hopf theorem.

Allées in Landscape Architecture and Garden Art—Types, Preservation, and Renewal of the Living Heritage of Baroque Allées in Hungary

Examining the history of garden art since ancient times, we find many examples of linear tree layouts supporting orientation or being used for the purpose of composition. Allées gained special significance during the Baroque as dynamic and grandiose space-forming garden design elements. They mostly consist of trees of taxonomically similar species planted along a regular line equidistant from each other in single or multiple rows. The two-dimensional compositional elements of the layout form three-dimensional longitudinal space forms. During their evolution, both their proportions and openness constantly change. Analyses of the compositional role and functions of allées of exemplary Hungarian Baroque garden complexes in the 18th century provided a basis for setting up a novel typology. Five compositional types have been defined as the primary result of archival research. The significance of the still-subsisting historic Hungarian allées calls for unique protection similar to European heritage protection. Taking a summary of significant, surviving examples of Hungarian Baroque allées into account, methods for allée renewal are defined along with the core question of whether allées are natural landscape elements or strict architectural compositions where authenticity may be an important criterion. The methodological research is partially based on three plans for the renewal of Baroque allées in Hungary that have been worked out by the Author as the chief landscape architect of the projects.

Classification of $$\eta $$ η -Biharmonic Surfaces in Non-flat Lorentz Space Forms

In this paper, we prove that $$\eta $$ -biharmonic surfaces in non-flat three-dimensional Lorentz space forms are isoparametric and give full classification results.

Soliton curves in three-dimensional space forms

The localized induction hierarchy is the integrable hierarchy for the localized induction equation, a geometric evolution equation for space curves modeled on the motion of vortex filaments. A solution of the n th stationary equation associated to the localized induction hierarchy is called an n th soliton curve. In this paper, we consider soliton curves in simply-connected three-dimensional space forms. In particular, we give explicit expressions of n th soliton curves in terms of the natural curvatures.

The complete classification of a class of conformally flat Lorentzian hypersurfaces in ℝ14

A three-dimensional Lorentzian hypersurface [Formula: see text] is called conformally flat if its induced metric is conformal to the flat Lorentzian metric, this property is preserved under the conformal transformation of [Formula: see text]. In this paper, using the projective light-cone model, we give a complete classification of those ones whose shape operators have two distinct real eigenvalues and cannot be diagonalizable. These hypersurfaces are conformal equivalent to cones, cylinders, or rotational hypersurfaces generated by B-scrolls (over null Frenet curves) in three-dimensional Lorentzian space forms.

Integrability aspects of the vortex filament equation for pseudo-null curves

An algebraic background in order to study the integrability properties of pseudo-null curve motions in a three-dimensional Lorentzian space form is developed. As an application, we delve into the relationship between the Burgers’ equation and the pseudo-null vortex filament equation. A recursion operator for the pseudo-null vortex filament equation is also provided.

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.

Surfaces with one constant principal curvature in three-dimensional space forms

We study surfaces with one constant principal curvature in Riemannian and Lorentzian three-dimensional space forms. Away from umbilic points they are characterized as one-parameter foliations by curves of constant curvature, each of them being centered at a point of a regular curve and contained in its normal plane. In some cases, a kind of trichotomy phenomenon is observed: the curves of the foliations may be circles, hyperbolas or horocycles, depending of whether the constant principal curvature is respectively larger, smaller of equal to one (not necessarily in this order). We describe explicitly some examples showing that there do exist complete surfaces with one constant principal curvature enjoying both umbilic and non-umbilic points.

CONGRUENCE SOLUTIONS TO THE LOCALIZED INDUCTION HIERARCHY IN THREE-DIMENSIONAL SPACE FORMS

The localized induction hierarchy in three-dimensional space forms is studied. In particular, we determine all the generating curves of congruence solutions to each evolution equation belonging to the localized induction hierarchy. Here, a congruence solution means a solution moving without changing shape. Also, we give the characterization of some low-order soliton curves.

Analysis on Structure and Morphological of Carton Packaging

The package structure is a three-dimensional space form, so people know the products are in used in the packaging. In packaging materials for paper use is very extensive, paper products are easy to shape the body shape for easy printing and recyclable advantage. This paper study design of the paper packaging structural, combining paper packaging structural design applications to explore the paper packaging structural morphology and environmentalist design consciousness.

Surfaces in three-dimensional space forms with divergence-free stress-bienergy tensor

We introduce the notion of biconservative hypersurfaces, that is hypersurfaces with conservative stress-energy tensor with respect to the bienergy. We give the (local) classification of biconservative surfaces in three-dimensional space forms.

On three-dimensional generalized sasakian space –forms

The object of the present paper is to study locally j –symmetric three-dimensional generalized Sasakian space forms and such manifolds with Ricci semi symmetric, h -parallel Ricci tensor and cyclic Ricci tensor. Such space forms with non-null concircular vector field are also considered. Key words: Generalized sasakian space forms, cosymplectic, locally-j -symmetric, h -parallel ricci tensor, cyclic ricci tensor and non-null concircular vector field.

Quotients of gravitational instantons

A classification result for Ricci-flat anti-self-dual asymptotically locally Euclidean 4-manifolds is obtained: they are either hyperkähler (one of the gravitational instantons classified by Kronheimer), or they are a cyclic quotient of a Gibbons–Hawking space. The possible quotients are described in terms of the monopole set in \({\mathbb{R}^3}\) , and it is proved that every such quotient is actually Kähler. The fact that the Gibbons–Hawking spaces are the only gravitational instantons to admit isometric quotients is proved by examining the possible fundamental groups at infinity: most can be ruled out by the classification of three-dimensional spherical space form groups, and the rest are excluded by a computation of the Rohklin invariant (in one case) or the eta invariant (in the remaining family of cases) of the corresponding space forms.

Kirchhoff elastic rods in three-dimensional space forms

The Kirchhoff elastic rod is one of the mathematical models of thin elastic rods, and is characterized as a critical point of the energy functional obtained by adding the effect of twisting to the bending energy. In this paper, we investigate Kirchhoff elastic rods in three-dimensional space forms. In particular, we give explicit formulas of Kirchhoff elastic rods in the three-sphere and in the three-dimensional hyperbolic space in terms of Jacobi sn function and the elliptic integrals.

Generalized Weierstrass representations of surfaces with the constant Gauss curvature in pseudo-Riemannian three-dimensional space forms

We know that there exists the correspondence between one of the nonumbilical Riemannian surface with constant Gauss curvature K in M¯3(K¯)(K≠K¯), the nonumbilical spacelike or timelike surfaces with constant Gauss curvature K in M¯13(K¯)(K≠K¯), and a solution of one of the following differential equations: the sine-Gordon, sinh-Gordon, sine-laplace, sinh-laplace, and cosh-Gordon equations. In this paper, we consider the initial value problems of these equations and give the generalized Weierstrass representations of these surfaces that depend only on the initial values of these equations.