Minimal curves of constant torsion

The Griffiths formalism is applied to find constant torsion curves which are extremal for arclength with respect to variations preserving torsion, fixing the endpoints and the binormals at the endpoints. The critical curves are elastic rods of constant torsion, which are shown to not realize certain boundary conditions.

     

On curves of constant torsion I

We give an explicit construction of a closed curve with constant torsion and everywhere positive curvature. We also discuss the restrictions on closed curves of constant torsion when they are constrained to lie on convex surfaces.     

Space curves with positive torsion

Summary Space curves may be classified under various kinds of deformation. The following six kinds of deformation have been of special interest; namely first, second and third order homotopy and isotopy. (We say the deformation is k-th order if the first k derivatives remain independent during the deformation.) The first order homotopy classification of space curves may be accomplished using well-known methods of Whitney; there is only one class. The second and third order homotopy classification was done by Feldman[1] and Little[6], respectively. The first order isotopy classification of space curves is knot theory; a subject of its own. The second order isotopy classification has been done by W. F. Pohl (unpublished). Thus, aside from knot theory, the only remaining problem is the third order isotopy problem. In this paper we give a partial answer. Our result is partial because we must restrict the class of curves with which we are dealing; namely to curves with a ? twist ?. But it may well be that every curve does have a twist, in which case our restricted class of curves would be all curves and the classification would be complete. In addition we construct a curve of positive torsion with any preassigned self-linking number in any preassigned knot class; a question raised by W. F. Pohl. Entrata in Redazione il 4 settembre 1976.     

Nonlinear partial differential equations associated with binormal motions of constant torsion curves in Minkowski 3-space

We give three nonlinear partial differential equations which are associated with binormal motions of constant torsion curves in Minkowski 3-space. We also give B?cklund transformations for these equations, as well as for surfaces swept out by related moving curves. As applications, from some trivial binormal motions we construct some new binormal motions.     

Fundamental S-units in hyperelliptic fields and the torsion problem in Jacobians of hyperelliptic curves

In 2010, Platonov proposed a fundamentally new approach to the torsion problem in Jacobi varieties of hyperelliptic curves over the field of rational numbers. This new approach is based on the calculation of fundamental units in hyperelliptic fields. It was applied to prove the existence of torsion points of new orders. In the paper, the notion of the degree of an S-unit is introduced and a relationship between the degree of an S-unit and the order of the corresponding torsion point of the Jacobian of a hyperelliptic curve is established. A complete exposition of the new method and results obtained on the basis of this method is contained in [2].     

Three-webs with covariantly constant curvature and torsion tensors

In this paper, we study a special class of multidimensional 3-webs with covariantly constant curvature and torsion tensors. In the first part, we prove that 3-webs of the class belong to G-webs, i.e., there is a subfamily of adapted frames whose components of curvature and torsion tensors are constant. The structure of the homogeneous space G/H carrying the 3-web is described. Structure equations of the G-group are found. In the second part, we find structure equations of the W ^??-web and finite equations of some special web classes.     

Elastic curves on the sphere

This paper deals with the derivation of equations suitable for the computation of elastic curves on the sphere. To this end, equations for the main invariants of spherical elastic curves are given. A new method for solving geometrically constraint differential equations is used to compute the curves for given initial values. A classification of the fundamental forms of the curves is presented.     

Inequalities for curves of constant breadth

Some theorems on curves of constant breadth are proved, which permit the simple estimate of breadth and of the radii of circum-circle and incircle. Moreover, they can be used for construction and judgement of approximation of such curves.Dedicated to Prof. Dr. Helmut Karzel on his sixtieth birthday     

Boxes for curves of constant length

We prove that every arc of lengthL inE ⁿ lies in some hypercube of diagonalL and that every closed curve of lengthL lies in some hypercube of diagonalL/2. In the casen=2, we find the smallest rectangle that can accommodate every arc of lengthL and the smallest rectangle that can accommodate every closed curve of lengthL.     

Diophantine analysis and torsion on elliptic curves

In a recent paper of Bennett and the author, it was shown thatthe elliptic curve defined by y² = x³ + Ax + B, where A andB are integers, has no rational points of finite order if Ais sufficiently large relative to B (at least if one assumesthe abc Conjecture of Masser and Oesterlé). In the presentarticle we show, perhaps surprisingly, that the rational torsionon the above curve is also quite restricted if B is sufficientlylarge relative to A. In particular, we demonstrate that forany > 0 there is a constant c such that if A and B are integerssatisfying |B| > c |A|⁶⁺, then the elliptic curve definedabove has no rational torsion points, other than a possiblepoint of order 2 (again making use of the abc Conjecture insome cases). We then extend this by proving similar resultsfor elliptic curves admitting non-trivial -isogenies, ellipticcurves written in other forms, and elliptic curves over certainnumber fields. Curiously, the results on isogenies lead to twounexpected irrationality measures for certain algebraic numbers.     

Centroaffine space curves with constant curvatures and homogeneous surfaces

We study the geometric characters of a centroaffine space curve with vanishing centroaffine curvatures, and classify the centroaffine space curves with constant centroaffine curvatures, which are centroaffine homogeneous curves in ${\mathbb{R}^3}$ . Moreover, we can find a centroaffine homogeneous surface on which such a space curve lies.     

Plateau's problem for H-convex curves

Various uniqueness results and non-uniqueness results for minimal surfaces are derived together with applications to boundary estimates and to finiteness theorems.Reasearch supported in part by NSF Grant 9210790.