On the logarithmic potential defined for a Van Koch curve

This is a continuation of [1]. Under study are the differentiability properties of the logarithmic potential determined for some class of complex measures distributed on Van Koch’s curves. Unlike the classical case of regular curves, the potential is shown to be of class C ¹ on the whole plane ℂ. We also study a related analog of Robin’s problem. The proofs are based on some results of [1].     

Survey curves for ovals

For a C¹-smooth strictly convex oval, the class of survey curves is introduced. The equivalence of four definitions of this class of curves is proved. The form of the shortest survey curve for an arbitrary oval is indicated and possible types of shortest survey curves are studied. Several external problems in the class of survey curves for the circle are solved. Bibliography: 4 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 246, 1997, pp. 13–35. Translated by N. Yu. Netsvetaev.     

Integrable (2+1)-dimensional systems of hydrodynamic type

We describe the results that have so far been obtained in the classification problem for integrable (2+1)-dimensional systems of hydrodynamic type. The Gibbons-Tsarev (GT) systems are most fundamental here. A whole class of integrable (2+1)-dimensional models is related to each such system. We present the known GT systems related to algebraic curves of genus g = 0 and g = 1 and also a new GT system corresponding to algebraic curves of genus g = 2. We construct a wide class of integrable models generated by the simplest GT system, which was not considered previously because it is “trivial.”     

Drinfeld Modules of Rank 1 and Algebraic Curves with Many Rational Points

 We construct algebraic curves C defined over a finite prime field such that the number of -rational points of C is large relative to the genus of C. The methods of construction are based on the relationship between algebraic curves and their function fields, as well as on narrow ray class extensions obtained from Drinfeld modules of rank 1. Received 21 July 1997; in revised form 5 February 1998     

Generalized planar curves and quaternionic geometry

Motivated by the analogies between the projective and the almost quaternionic geometries, we first study the generalized planar curves and mappings. We follow, recover, and extend the classical approach, see e.g., (Sov. Math. 27(1) 63–70 (1983), Rediconti del circolo matematico di Palermo, Serie II, Suppl. 54 75–81) (1998), Then we exploit the impact of the general results in the almost quaternionic geometry. In particular we show, that the natural class of ℍ-planar curves coincides with the class of all geodesics of the so called Weyl connections and preserving this class turns out to be the necessary and sufficient condition on diffeomorphisms to become morphisms of almost quaternionic geometries.     

Laminar currents in ℙ<Superscript>2</Superscript>

 In this paper we study laminar currents in ℙ². Given a sequence of irreducible algebraic curves (C _n ) converging in the sense of currents to T, we find geometric conditions on the curves ensuring that the limit current T is laminar. This criterion is then applied to meromorphic dynamical systems in ℙ², and laminarity of the dynamical ``Green' current is obtained for a wide class of meromorphic self maps of ℙ², as well as for all bimeromorphic maps of projective surfaces. Received: 24 September 2001 / Published online: 10 February 2003 Mathematics Subject Classification (2000): 32U40, 37Fxx, 32H50     

Algebraic families of smooth hyperbolic surfaces of low degree in ℙ ℂ 3

We reprove (after a paper of Y.T. Siu appeared in 1987) a simple vanishing theorem for the Wronskian of Brody curves under a suitable assumption on the existence of global meromorphic connections. Next we give a slight improvement of a result due to Y.T, Siu and A.M. Nadel (Duke Math. J., 1989) on the algebraic degeneracy of entire holomorphic curves contained in certain hypersurfaces of ℙ _ℂ ⁿ . Especially, their result is generalized to a larger class of hypersurfaces. Our method produces algebraic families of smooth hyperbolic surfaces in ℙ _ℂ ³ for all degreesd≥14; this brings us somewhat nearer than previously known from the expected ranged≥5.     

Drinfeld Modules of Rank 1 and Algebraic Curves with Many Rational Points

 We construct algebraic curves C defined over a finite prime field such that the number of -rational points of C is large relative to the genus of C. The methods of construction are based on the relationship between algebraic curves and their function fields, as well as on narrow ray class extensions obtained from Drinfeld modules of rank 1.     

Real structures of Teichmüller spaces, Dehn twists, and moduli spaces of real curves

An orientation reversing involution of a topological compact genus surface induces an antiholomorphic involution of the Teichmüller space of genus g Riemann surfaces. Two such involutions and are conjugate in the mapping class group if and only if the corresponding orientation reversing involutions and of are conjugate in the automorphism group of . This is equivalent to saying that the quotient surfaces and are homeomorphic. Hence the Teichmüller space has distinct antiholomorphic involutions, which are also called real structures of ([7]). This result is a simple fact that follows from Royden's theorem ([4]) stating that the the mapping class group is the full group of holomorphic automorphisms of the Teichmüller space (). Let and be two real structures that are not conjugate in the mapping class group. In this paper we construct a real analytic diffeomorphism such that This mapping d is a product of full and half Dehn–twists around certain simple closed curves on the surface . This has applications to the moduli spaces of real algebraic curves. A compact Riemann surface admitting an antiholomorphic involution is a real algebraic curve of the topological type . All fixed–points of the real structure of the Teichmüller space , are real curves of the above topological type and every real curve of that topological type is represented by an element of the fixed–point set of . The fixed–point set is the Teichmüller space of real algebraic curves of the corresponding topological type. Given two different real structures and , let d the the real analytic mapping satisfying (1). It follows that d maps onto and is an explicit real analytic diffeomorphism between these Teichmüller spaces. Received 8 December 1997; accepted 12 August 1998     

Counting curves on rational surfaces

In [CH3], Caporaso and Harris derive recursive formulas counting nodal plane curves of degree d and geometric genus g in the plane (through the appropriate number of fixed general points). We rephrase their arguments in the language of maps, and extend them to other rational surfaces, and other specified intersections with a divisor. As applications, (i) we count irreducible curves on Hirzebruch surfaces in a fixed divisor class and of fixed geometric genus, (ii) we compute the higher-genus Gromov–Witten invariants of (or equivalently, counting curves of any genus and divisor class on) del Pezzo surfaces of degree at least 3. In the case of the cubic surface in (ii), we first use a result of Graber to enumeratively interpret higher-genus Gromov–Witten invariants of certain K-nef surfaces, and then apply this to a degeneration of a cubic surface. Received: 30 June 1999 / Revised version: 1 January 2000     

Coexistence of small and large amplitude limit cycles of polynomial differential systems of degree four

A class of degree four differential systems that have an invariant conic x ² + Cy ² = 1, C ∈ ℝ, is examined. We show the coexistence of small amplitude limit cycles, large amplitude limit cycles, and invariant algebraic curves under perturbations of the coefficients of the systems.     

Pythagorean-hodograph space curves

We investigate the properties of polynomial space curvesr(t)={x(t), y(t), z(t)} whose hodographs (derivatives) satisfy the Pythagorean conditionx′²(t)+y′²(t)+z′²(t)≡σ²(t) for some real polynomial σ(t). The algebraic structure of thecomplete set of regular Pythagorean-hodograph curves in ℝ³ is inherently more complicated than that of the corresponding set in ℝ². We derive a characterization for allcubic Pythagoreanhodograph space curves, in terms of constraints on the Bézier control polygon, and show that such curves correspond geometrically to a family of non-circular helices. Pythagorean-hodograph space curves of higher degree exhibit greater shape flexibility (the quintics, for example, satisfy the general first-order Hermite interpolation problem in ℝ³), but they have no “simple” all-encompassing characterization. We focus on asubset of these higher-order curves that admits a straightforward constructive representation. As distinct from polynomial space curves in general, Pythagorean-hodograph space curves have the following attractive attributes: (i) the arc length of any segment can be determined exactly without numerical quadrature; and (ii) thecanal surfaces based on such curves as spines have precise rational parameterizations.     

Hausdorff diameter of a nonsymmetric class of vector functions

In the space of parametrically determined m-dimensional curves with the Hausdorff metric, we find the diameter of a class of curves whose coordinate functions satisfy the Lipschitz condition on some segment and take fixed values at its endpoints. We obtain the dependence of relations determining the value of the diameter on the evenness of m. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 11, pp. 1570–1573, November, 1998.     

Length Bounds on Curves Arising from Tight Geodesics

Let M be a complete hyperbolic 3-manifold admitting a homotopy equivalence to a compact surface ∑, such that the cusps of M are in bijective correspondence with the boundary components of ∑. Suppose we realise a tight geodesic in the curve complex as a sequence of closed geodesics M. There is an upper bound on the lengths of such curves in terms of the lengths of the terminal curves and the topologicial type of ∑. We give proofs of these and related bounds. Similar bounds have been proven by Minsky using the sophisticated machinery of hierarchies. Such bounds feature in the work of Brock, Canary and Minsky towards the ending lamination conjecture, and can also be used to study the action of the mapping class group on the curve complex. Received: January 2006, Revision: March 2007, Accepted: July 2007     

Global curvature for surfaces and area minimization under a thickness constraint

Motivated by previous work on elastic rods with self-contact, involving the concept of the global radius of curvature for curves (as defined by Gonzalez and Maddocks), we define the global radius of curvature Δ[X] for a wide class of continuous parametric surfaces X for which the tangent plane exists on a dense set of parameters. It turns out that in this class of surfaces a positive lower bound Δ[X] ≥ θ > 0 provides, naively speaking, the surface with a thickness of magnitude θ; it serves as an excluded volume constraint for X, prevents self-intersections, and implies that the image of X is an embedded C¹-manifold with a Lipschitz continuous normal. We also obtain a convergence and a compactness result for such thick surfaces, and show one possible application to variational problems for embedded objects: the existence of ideal surfaces of fixed genus in each isotopy class. The proofs are based on a mixture of elementary topological, geometric and analytic arguments, combined with a notion of the reach of a set, introduced by Federer in 1959. Mathematics Subject Classification (2000) 49Q10, 53A05, 53C45, 57R52, 74K15     

On the Rao modules of minimal curves in $${\mathbb{P}^N}$$

We study the Hartshorne-Rao modules M _C of minimal curves C in \mathbbP^N{\mathbb{P}^N} , with N ≥ 4, lying in the same liaison class of curves on a smooth rational scroll surface. We get a free minimal resolution of M _C for some of such curves and an upper bound for Betti numbers of M _C , for any C.     

An algorithm for finding planar surfaces in three-manifolds

By a slope in the boundary ∂M of a 3-manifold, we mean the isotopy class α of a finite set of disjoint simple closed curves in ∂M that are nontrivial and pairwise nonparallel. In this paper, we construct an algorithm to decide whether or not a given orientable 3-manifold M contains an essential planar surface whose boundary has a given slope α. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 4, pp. 197–202, 2005.     

Donaldson theory on non-Kählerian surfaces and class <Emphasis Type="Italic">VII</Emphasis> surfaces with <Emphasis Type="Italic">b</Emphasis><Subscript>2</Subscript>=1

We prove that any class VII surface with b₂=1 has curves. This implies the “Global Spherical Shell conjecture” in the case b₂=1: Any minimal class VII surface withb₂=1 admits a global spherical shell, hence it is isomorphic to one of the surfaces in the known list. By the results in [LYZ], [Te1], which treat the case b₂=0 and give complete proofs of Bogomolov’s theorem, one has a complete classification of all class VII-surfaces with b₂∈{0,1}. The main idea of the proof is to show that a certain moduli space of PU(2)-instantons on a surface X with no curves (if such a surface existed) would contain a closed Riemann surface Y whose general points correspond to non-filtrable holomorphic bundles on X. Then we pass from a family of bundles on X parameterized by Y to a family of bundles on Y parameterized by X, and we use the algebraicity of Y to obtain a contradiction. The proof uses essentially techniques from Donaldson theory: compactness theorems for moduli spaces of PU(2)-instantons and the Kobayashi-Hitchin correspondence on surfaces.     

Implementing the 4-dimensional GLV method on GLS elliptic curves with <Emphasis Type="Italic">j</Emphasis>-invariant 0

The Gallant–Lambert–Vanstone (GLV) method is a very efficient technique for accelerating point multiplication on elliptic curves with efficiently computable endomorphisms. Galbraith et al. (J Cryptol 24(3):446–469, 2011) showed that point multiplication exploiting the 2-dimensional GLV method on a large class of curves over \mathbbF_p²{\mathbb{F}_{p^2}} was faster than the standard method on general elliptic curves over \mathbbF_p{\mathbb{F}_{p}} , and left as an open problem to study the case of 4-dimensional GLV on special curves (e.g., j (E) = 0) over \mathbbF_p²{\mathbb{F}_{p^2}} . We study the above problem in this paper. We show how to get the 4-dimensional GLV decomposition with proper decomposed coefficients, and thus reduce the number of doublings for point multiplication on these curves to only a quarter. The resulting implementation shows that the 4-dimensional GLV method on a GLS curve runs in about 0.78 the time of the 2-dimensional GLV method on the same curve and in between 0.78 − 0.87 the time of the 2-dimensional GLV method using the standard method over \mathbbF_p{\mathbb{F}_{p}} . In particular, our implementation reduces by up to 27% the time of the previously fastest implementation of point multiplication on x86-64 processors due to Longa and Gebotys (CHES2010).     

The index of singular integral operators in reflexive Orlicz spaces

We consider the Banach algebra of singular integral operators with matrix piecewise continuous coefficients in the reflexive Orlicz spaceL _M ⁿ (Γ). We assume that Γ belongs to a certain wide subclass of the class of Carleson curves; this subclass includes curves with cusps, as well as curves of the logarithmic spiral type. We obtain an index formula for an arbitrary operator from the algebra in terms of the symbol of this operator. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 383–396, September, 1998.