Compact Curves Research Articles

Wavelet Multiresolution Representation of Curves and Surfaces

We develop wavelet methods for the multiresolution representation of parametric curves and surfaces. To support the representation, we construct a new family of compactly supported symmetric biorthogonal wavelets with interpolating scaling functions. The wavelets in these biorthogonal pairs have properties better suited for curves and surfaces than many commonly used filters. We also give examples of the applications of the wavelet approach: these include the derivation of compact hierarchical curve and surface representations using modified wavelet compression, the identification of smooth sections of surfaces, and a subdivision-like intersection algorithm for discrete plane curves.

On the persistence of pseudo-holomorphic curves on an almost complex torus (with an appendix by J�rgen P�schel)

In recent years pseudo-holomorphic curves on almost complex manifolds have played an important role in connection with the developments in symplectic geometry, notably on account of Gromov's work [8] in 1985. In particular, the existence of compact pseudo-holomorphic curves of the type of S 2 or the closed disc have been established on particular manifolds. In this paper we study, unrelated from symplectic geometry, non-compact pseudo-holomorphic curves on an almost complex torus (T2n,J). These curves carry a complex structure and therefore represent Riemann surfaces. In our case they are of the type of ~ or a cylinder ~* = IE\(0). In particular we are interested in the question of the persistence of these curves under perturbation of the almost complex structure. This question has similar features as that of the persistence of invariant tori in classical mechanics, as initiated by Kolmogorov [2,15,18]. However, in contrast with that theory we are dealing with nonlinear elliptic systems of partial differential equations of Cauchy-Riemann type. Our results provide global solutions for such systems.

Characterization of smooth, compact algebraic curves in ℝ²

Characterization of smooth, compact algebraic curves in ℝ²

Indices of holomorphic vector fields relative to invariant curves on surfaces

The local index of a holomorphic vector field relative to a (possibly singular) invariant complex analytic curve on a complex surface is defined and it is shown that, for a compact curve invariant by a one-dimensional singular foliation on a surface, the sum of the indices is equal to its self-intersection number. An interpretation of the indices in terms of holonomy is also given.

Universal central extensions of elliptic affine Lie algebras

Let 𝔤 be a simple complex (finite dimensional) Lie algebra, and let R be the ring of regular functions on a compact complex algebraic curve with a finite number of points removed. Lie algebras of the form 𝔤⊗CR are considered; these generalize Kac–Moody loop algebras since for a curve of genus zero with two punctures R≂C[t,t−1]. The universal central extension of 𝔤⊗R is analogous to an untwisted affine Kac–Moody algebra. By Kassel’s theorem the kernel of the universal central extension is linearly isomorphic to the Kähler differentials of R modulo exact differentials. The dimension of the kernel for any R is determined first. Restricting to hyperelliptic curves with 2, 3, or 4 special points removed, a basis for the kernel is determined. Restricting further to an elliptic curve with punctures at two points (of orders one and two in the group law) we explicitly determine the cocycles which give the commutation relations for the universal central extension. The results involve Pollaczek polynomials, which are a genus-one generalization of ultraspherical (Gegenbauer) polynomials. © 1994 American Institute of Physics.

Interrelationships between mixing ratios of long‐lived stratospheric constituents

Recent analyses have revealed simple relationships between the simultaneously measured mixing ratios of certain stratospheric constituents. In some cases, the relationship appears to be nearly linear, so that measured concentrations of one can be used to predict the other. We argue here that such relationships are to be expected for species of sufficiently long lifetime. Species whose local lifetimes are longer than quasi‐horizontal transport time scales are in climatological slope equilibrium, i.e., they share surfaces of constant mixing ratio, and a scatter plot of the mixing ratio of one versus that of the other collapses to a compact curve whose slope at any point is the ratio of the net global fluxes of the two species through the corresponding surface of constant mixing ratio. Species whose local lifetimes are greater than vertical transport time scales are in gradient equilibrium and their mixing ratios display a linear relationship. For species whose atmospheric lifetimes are determined by removal in the stratosphere, the slope of this relationship in the lower stratosphere can be related to the ratio of their atmospheric lifetimes. These statements are illustrated using results from a two‐dimensional chemistry‐transport model.

Holomorphic curves in surfaces of general type.

This note answers some questions on holomorphic curves and their distribution in an algebraic surface of positive index. More specifically, we exploit the existence of natural negatively curved "pseudo-Finsler" metrics on a surface S of general type whose Chern numbers satisfy c(2)1>2c2 to show that a holomorphic map of a Riemann surface to S whose image is not in any rational or elliptic curve must satisfy a distance decreasing property with respect to these metrics. We show as a consequence that such a map extends over isolated punctures. So assuming that the Riemann surface is obtained from a compact one of genus q by removing a finite number of points, then the map is actually algebraic and defines a compact holomorphic curve in S. Furthermore, the degree of the curve with respect to a fixed polarization is shown to be bounded above by a multiple of q - 1 irrespective of the map.

Dimension and structure of typical compact sets, continua and curves

Most compact sets have small Hausdorff and lower entropy dimension, large upper entropy dimension, each of their points is “almost isolated” and they are “very porous”. Similar statements hold for continua, continuous curves and graphs of real continuous functions.

Decay rates of Fourier transforms of curves

Let d μ d\mu be a smooth measure on a nondegenerate curve in R n {{\mathbf {R}}^n} . This paper examines the decay rate of spherical averages of its Fourier transform d μ ^ \widehat {d\mu } . Thus estimates of the following form are considered: \[ ( ∫ ∑ r | d μ ^ ( ξ | p d ξ ) 1 / p ⩽ C r − σ | | f | | {\left ( {\int _{{\sum _r}} {|\widehat {d\mu }(\xi {|^p}d\xi } } \right )^{1/p}} \leqslant C{r^{ - \sigma }}||f|| \] where ∑ r = { ξ ∈ R n : | ξ | = r } {\sum _r} = \{ \xi \in {{\mathbf {R}}^n}:|\xi | = r\} .

Horizontal holomorphic curves in 𝑆𝑝(𝑛)-flag manifolds

A complete description of horizontal holomorphic curves in Sp ( n ) {\text {Sp}}\left ( n \right ) -flag manifolds is given. For compact curves a Plücker type integral formula is derived.

Frenet formulae for holomorphic curves in the two quadric

We give a complete description of holomorphic curves in the complex two quadric via the method of moving frames. For compact curves a Morse theory type integral formula is derived.

The Asymptotic Behavior of a Variation of Polarized Hodge Structure

0.1. The purpose of this paper is to give the asymptotic behavior of variation of polarized Hodge structures in the several-dimensional case. We do not discuss here why and how the notion of variation of Hodge structures arises and it is developed by P. A. Griffiths, P. Deligne, W. Schmid and others. What motivates us is to generalize Zucker's result to the several-dimensional case. His result is as follows: the cohomology groups of a variation of Hodge structure on the compact curve with finite singular points have also a Hodge structure,, He proceeds his proof as follows. As an analytic tool, he uses the harmonic analysis (Hodge-Kodaira theory) and as a geometric tool he uses W. Schmid's result that we discuss later. By using the Kahler metric on the curve which behaves with the special property at singular points and the Hermitian metric of the vector bundle which arises from the polarization of Hodge structure, he succeeds to express the cohomology groups of the variation as the L-cohomology groups. Since the L-cohomology group is isomorphic to the space of harmonic forms, by decomposing harmonic forms into (/?, q} -forms, he obtains the Hodge decomposition of the cohomology group of Hodge structure. However, in order to prove the first step—to express the cohomology group by L-cohomology group—he is obliged to use the result of W. Schmid on the asymptotic behavior of variations of Hodge structures at singularity. In this paper, we generalize W. Schmid's result to the several-

On Weakly 1-Complete Surfaces without Non-Constant Holomorphic Functions

In this paper, we are interested in a class of two dimensional complex manifolds which are called weakly 1-complete surfaces. Here we call a two dimensional complex manifold X a weakly 1-complete surface if X possesses a C°°-exhausting plurisubharmonic function. This class includes two different extreme objects: compact analytic surfaces and two dimensional Stein manifolds. But at the same time, this class includes some curious examples from the function theoretic point of view i.e. there are weakly 1-complete surfaces without non-constant holomorphic functions (see [3] [6] [8] [12]) and moreover non-compact weakly 1-complete surfaces have an extreme function theoretic property i.e. a non-compact weakly 1-complete surface X is holomorphically convex if and only if X possesses a non-constant holomorphic function (see [9]). Looking back to the case of compact analytic surfaces, roughly speaking, they are classified by the existence or non-existence of meromorphic function. Hence it is natural to suppose that this aspect might give a new standpoint to analyze such a curious example in the class of weakly 1-complete surfaces as far as weakly 1-completeness is expected as a nice intermediate concept between compactness and Stein. This note is an attempt towards the problem of the existence of meromorphic function on non-compact weakly 1-complete surfaces. From now on, all weakly 1-complete surfaces are connected and non-compact and have no exceptional compact curves of the first kind unless otherwise is explicitly stated. Then we shall prove the following theorem.

On the neighborhood of a compact complex curve with topologically trivial normal bundle

On the neighborhood of a compact complex curve with topologically trivial normal bundle

The theorem of Torelli for singular curves

Let C be a compact (singular) curve embedded in a surface. Then C carries a canonical sheaf Ω \Omega which is locally free of rank 1. Moreover, C has a generalized Jacobian J which fits in an exact sequence ( ∗ ) 0 → F → J → A → 0 \begin{equation}\tag {$\ast $} 0 \to F \to J \to A \to 0\end{equation} of algebraic groups such that A is an abelian variety and F = ( C ∗ ) r × C s F = {({{\mathbf {C}}^\ast })^r} \times {{\mathbf {C}}^s} . Let C _ \underline {C} be the set of nonsingular points of C and let θ \theta = Zariski-closure of the image of ( C _ ) ( g − 1 ) (\underline {C})^{(g - 1)} in J. Then: Theorem. If C is irreducible and sections of Ω \Omega map C onto X in P g − 1 {P^{g - 1}} then the isomorphism class of J together with the translation class of the divisor θ \theta on J determine the isomorphism class of X. As a corollary, if ψ : C → X \psi :C \to X is an isomorphism (in which case we call C nonhyperelliptic) the above data determine the isomorphism class of C. I do not know if this remains true when C is hyperelliptic. It should be noted that the linear equivalence class of θ \theta is not enough to determine X. The principal idea of the proof is that of Andreotti, that is, to recover the curve as the dual of the branch locus of the Gauss map from θ \theta to P g − 1 {P^{g - 1}} ; however our arguments are usually analytic. The organization of this paper is as follows: In §1 we prove a stronger than usual version of Abel’s theorem for Riemann surfaces and in §2 we extend this theorem to apply to singular curves. In succeeding sections we construct the generalized Jacobian as a complex Lie group J and embed J in an analytic fibre bundle over A with projective spaces as fibre. This we use to endow J with the structure of an algebraic group. §7 contains a miscellany of facts about branch loci and dual varieties, and in §8 the main theorems are stated and proved. We should mention here that the variations on Abel’s theorem proved in this paper (1.2.4 and 3.0.1) were proved by Severi, at least in the special case corresponding to ordinary double points [12].

Neighborhoods of a Compact Non-Singular Algebraic Curve Imbedded in a 2-Dimensional Complex Manifold

Let C be a compact non-singular algebraic curve imbedded in a 2-dimensional complex manifold S. In this paper we consider the following problem: Under what conditions does there exist a non-constant holomorphic function defined on a small neighborhood of Cl The signature of the normal bundle Nc is not sufficient to solve our problem (see, Table in § 6 and Theorem 2 in § 5). Hence we have to introduce the concept of a regularly half pseudoconvex neighborhood system of C (for definition, see (1.1)). Then the necessary and sufficient condition is given in the following

One-Dimensional Continuous Curves and A Homogeneity Theorem

Let M be the class of all one-dimensional compact metric locally connected continua up to topological equivalence. Let SU be the subclass of M consisting of those elements of M having no local cut points2 Each of the universal curve and the universal plane curve' is an element of WL. One of the principal results of this paper asserts that if an element M of JJ contains no open subset imbeddable in the plane then M is the universal curve. Thus we have a simple characterization of the universal curve. As a corollary of this result and known theorems, it follows that an element K of M is homogeneous if and only if K is a simple closed curve or a universal curve.4 In other words, there are exactly two homogeneous one-dimensional compact continuous curves. A further characterization of the universal curve is also given, as are a number of corollaries of the characterizations. Generalizations to the locally compact, non-compact cases are included in Section 6.

A note on continuous collections of continuous curves filling up a continuous curve in the plane

This theorem provides a complement to Theorem VI of [1] which states that if G is a continuous collection of nondegenerate compact continuous curves in the plane which is a compact continuum with respect to its elements as points, then G is a hereditary continuous curve such that the closure of its set of emanation points is totally disconnected. Some notation similar to that of [1 ] will be used. A simple chain is a finite collection x1, x2, * * *, xn of open discs (interiors of simple closed curves) such that xi x; exists if and only if I i-il < 1 and is a closed disc (a simple closed curve plus its interior) if it does exist. A (simple) chain C is said to simply cover a set M if C* contains M but for no proper subchain C' of C does the closure of C'* contain M.' Two chains, C and C', are said to be mutually exclusive if C* and C'* are mutually exclusive. The chain C is said to be a closed refinement of the chain C' provided that each link of C' contains the closure of some link of C and the closure of each link of C is contained in some link of C'. PROOF OF THEOREM. We note first that only a countable number of the elements of G contain triods. (See [5].) Denote the elements of some subset of G containing these elements by gl, g2, * C C . Each of the remaining elements of G is an arc or a simple closed curve. If G is not an arc or a simple closed curve with respect to its elements as points, it has an emanation element. Let H denote the closure of the set of emanation elements of G. It has been noted that, by Theorem VI of [1], H is totally disconnected. Let T be an arc of elements of G and suppose that the non-endelement t of T is an isolated element of T H. There is a subarc T1 of

Continuous collections of continuous curves in the plane

The purpose of this paper is to consider the class of continuous collections of mutually exclusive compact continuous curves in the plane. Throughout this paper we shall denote by G or G with a subscript or superscript a collection of this class with the property that G with respect to its elements as points is a nondegenerate compact closed point set. G has then the significance of being both a collection of continua and a point set itself. By a continuous curve will be meant a nondegenerate locally connected compact continuum. By a continuous collection will be meant a collection which is both upper and lower semi-continuous. By a (simple) chain will be meant a finite collection xi, x2, * *, x. of open discs (i.e. interiors of simple closed curves) such that i,.j exists if and only if j i-ijI ?1 and is the closure of an open disc (i.e. a 2-cell) if it does exist. The xi are called links of the chain. A subchain of a chain c is a chain whose links are links of c. A chain c will be said to simply cover a set M if c* contains M and if for no proper subchain c' of c does the closure of c'* contain M. Two chains will be said to be mutually exclusive if no link of either intersects any link of the other. A collection C' of sets is said to be a (closed) refinement of a collection C of sets if (the closure of) each element of C' is a subset of some element of C. An emanation point of a continuum M is a point which is the common part of each pair of some three nondegenerate subcontinua of M. A hereditary continuous curve is a continuous curve each of whose nondegenerate subcontinua is a continuous curve. It is immediately clear that if G is connected, G contains uncountably many elements and that only countably many can contain triods [1]. Except for a countable number of elements, each element of G must be either an arc or a simple closed curve. We denote the elements of G which are neither arcs nor simple closed curves by g1, g2, g3, * . . From the hypothesis of continuity of G it follows immediately that no element of a connected G contains a 2-cell.

Concerning upper semi-continuous collections of continua

Hurewicz [1](2) and Mazurkiewicz [2] showed independently that if M is any compact metric continuum, there exist a one-dimensional continuum K in three-dimensional Euclidean space and an upper semi-continuous collection [3 ] of mutually exclusive continua filling up K which with respect to its elements as points is topologically equivalent to M. In the case of each of these solutions the method of proof used does not lend itself readily to the solution of the principal result of this paper which is the demonstration that if M is any compact continuous curve, there exist a one-dimensional continuous curve K in three-dimensional Euclidean space and an upper semi-continuous collection of mutually exclusive continua filling up K which with respect to its elements as points is topologically equivalent to M. It is known that, in the problem of Hurewicz and Mazurkiewicz, if M is not a continuous curve then K cannot be a continuous curve. The principal result in this paper, Theorem II, was proposed to me as a problem by Professor R. L. Moore. I wish to express my sincere appreciation to Professor Moore for his patient and stimulating teaching and for his contagious enthusiasm for mathematical research. DEFINITION. A collection Q of continuous curves will be said to have the X property if the common part of the continua of any subcollection of Q has only a finite number of components, each a continuous curve.