Compact Curves Research Articles

Relative K-polystability of projective bundles over a curve

Let $P(E)$ be the projectivization of a holomorphic vector bundle $E$ over a compact complex curve $C$. We characterize the existence of an extremal K\ahler metric on the ruled manifold $P(E)$ in terms of relative K-polystability and the fact that $E$ decomposes as a direct sum of stable bundles.

Complex Geometry of Iwasawa Manifolds

Abstract An Iwasawa manifold is a compact complex homogeneous manifold isomorphic to a quotient $G/\Lambda $, where $G$ is the group of complex unipotent $3 \times 3$ matrices and $\Lambda \subset G$ is a cocompact lattice. In this work, we study holomorphic submanifolds in Iwasawa manifolds. We prove that any compact complex curve in an Iwasawa manifold is contained in a holomorphic subtorus. We also prove that any complex surface in an Iwasawa manifold is either an abelian surface or a Kähler non-projective isotrivial elliptic surface of Kodaira dimension one. In the Appendix, we show that any subtorus in Iwasawa manifold carries complex multiplication.

Conformal field theories and compact curves in moduli spaces

We show that there are many compact subsets of the moduli space Mg of Riemann surfaces of genus g that do not intersect any symmetry locus. This has interesting implications for mathcal{N}=2 supersymmetric conformal field theories in four dimensions.

Abel Maps and Limit Linear Series for Curves of Compact Type with Three Irreducible Components

We explore the relationship between limit linear series and fibers of Abel maps for compact type curves with three components. For compact type curves with two components, given an exact Osserman limit linear series $${\mathfrak {g}}$$ , Esteves and Osserman associated a closed subscheme $${\mathbb {P}}({\mathfrak {g}})$$ of the fiber of the corresponding Abel map. We generalize this definition to our case. Then, for $${\mathfrak {g}}$$ the unique exact extension of an r-dimensional refined Eisenbud–Harris limit linear series, we find the irreducible components of $${\mathbb {P}}({\mathfrak {g}})$$ and we show that $${\mathbb {P}}({\mathfrak {g}})$$ is connected of pure dimension r, with the same Hilbert polynomial as the diagonal in $${\mathbb {P}}^{r}\times {\mathbb {P}}^{r}\times {\mathbb {P}}^{r}$$ .

Quantification of Influence of Epoxy Layer Between Interface Steel-Concrete on Calibration Curves for Modified Compact Tension Test

AbstractThe objective of this study is to investigate the use of compact tension specimen calibration curves for evaluation data of fracture properties measurement performed on modified compact tension specimen made from cement based composites. From literature, the well-known calibration curve for compact tension specimen is compared with calibration curves for Modified compact tension specimen that are obtained from numerical calculation. The obtained results are quantified and accuracy of solution is discussed. The suggested curves could be preferably used for determining of the fracture parameters.

Twistor theory at fifty: from contour integrals to twistor strings.

We review aspects of twistor theory, its aims and achievements spanning the last five decades. In the twistor approach, space-time is secondary with events being derived objects that correspond to compact holomorphic curves in a complex threefold-the twistor space. After giving an elementary construction of this space, we demonstrate how solutions to linear and nonlinear equations of mathematical physics-anti-self-duality equations on Yang-Mills or conformal curvature-can be encoded into twistor cohomology. These twistor correspondences yield explicit examples of Yang-Mills and gravitational instantons, which we review. They also underlie the twistor approach to integrability: the solitonic systems arise as symmetry reductions of anti-self-dual (ASD) Yang-Mills equations, and Einstein-Weyl dispersionless systems are reductions of ASD conformal equations. We then review the holomorphic string theories in twistor and ambitwistor spaces, and explain how these theories give rise to remarkable new formulae for the computation of quantum scattering amplitudes. Finally, we discuss the Newtonian limit of twistor theory and its possible role in Penrose's proposal for a role of gravity in quantum collapse of a wave function.

FlowRep

We present FlowRep , an algorithm for extracting descriptive compact 3D curve networks from meshes of free-form man-made shapes. We infer the desired compact curve network from complex 3D geometries by using a series of insights derived from perception, computer graphics, and design literature. These sources suggest that visually descriptive networks are cycle-descriptive , i.e their cycles unambiguously describe the geometry of the surface patches they surround. They also indicate that such networks are designed to be projectable , or easy to envision when observed from a static general viewpoint; in other words, 2D projections of the network should be strongly indicative of its 3D geometry. Research suggests that both properties are best achieved by using networks dominated by flowlines , surface curves aligned with principal curvature directions across anisotropic regions and strategically extended across sharp-features and isotropic areas. Our algorithm leverages these observation in the construction of a compact descriptive curve network. Starting with a curvature aligned quad dominant mesh we first extract sequences of mesh edges that form long, well-shaped and reliable flowlines by leveraging directional similarity between nearby meaningful flowline directions We then use a compact subset of the extracted flowlines and the model's sharp-feature, or trim, curves to form a sparse, projectable network which describes the underlying surface. We validate our method by demonstrating a range of networks computed from diverse inputs, using them for surface reconstruction, and showing extensive comparisons with prior work and artist generated networks.

Metric uniformization of morphisms of Berkovich curves

We show that the metric structure of morphisms f:Y→X between quasi-smooth compact Berkovich curves over an algebraically closed field admits a finite combinatorial description. In particular, for a large enough skeleton Γ=(ΓY,ΓX) of f, the sets Nf,≥n of points of Y of multiplicity at least n in the fiber are radial around ΓY with the radius changing piecewise monomially along ΓY. In this case, for any interval l=[z,y]⊂Y connecting a point z of type 1 to the skeleton, the restriction f|l gives rise to a profile piecewise monomial function φy:[0,1]→[0,1] that depends only on the type 2 point y∈ΓY. In particular, the metric structure of f is determined by Γ and the family of the profile functions {φy} with y∈ΓY(2). We prove that this family is piecewise monomial in y and naturally extends to the whole Y. In addition, we extend the classical theory of higher ramification groups to arbitrary real-valued fields and show that φy coincides with the Herbrand function of H(y)/H(f(y)). This gives a curious geometric interpretation of the Herbrand function, which also applies to non-normal and even inseparable extensions.

Some properties of Grauert type surfaces

In a previous work, we classified weakly complete surfaces which admit a real analytic plurisubharmonic exhaustion function; we showed that, if they are not proper over a Stein space, then they admit a pluriharmonic function, with compact Levi-flat level sets foliated with dense complex leaves. We called these Grauert type surfaces. In this note, we investigate some properties of these surfaces. Namely, we prove that the only compact curves that can be contained in them are negative in the sense of Grauert and that the level sets of the pluriharmonic function are connected, thus completing the analogy with the Cartan–Remmert reduction of a holomorphically convex space. Moreover, in our classification theorem, we had to pass to a double cover to produce the pluriharmonic function; the last part of the present paper is devoted to the construction of an example where passing to a double cover cannot be avoided.

Holomorphic Legendrian curves

In this paper we study holomorphic Legendrian curves in the standard holomorphic contact structure on$\mathbb{C}^{2n+1}$for any$n\in \mathbb{N}$. We provide several approximation and desingularization results which enable us to prove general existence theorems, settling some of the open problems in the subject. In particular, we show that every open Riemann surface$M$admits a proper holomorphic Legendrian embedding$M{\hookrightarrow}\mathbb{C}^{2n+1}$, and we prove that for every compact bordered Riemann surface$M={M\unicode[STIX]{x0030A}}\,\cup \,bM$there exists a topological embedding$M{\hookrightarrow}\mathbb{C}^{2n+1}$whose restriction to the interior is a complete holomorphic Legendrian embedding${M\unicode[STIX]{x0030A}}{\hookrightarrow}\mathbb{C}^{2n+1}$. As a consequence, we infer that every complex contact manifold$W$carries relatively compact holomorphic Legendrian curves, normalized by any given bordered Riemann surface, which are complete with respect to any Riemannian metric on$W$.

Quasiconformal mappings on the Grushin plane

We prove that a self-homeomorphism of the Grushin plane is quasisymmetric if and only if it is metrically quasiconformal and if and only if it is geometrically quasiconformal. As the main step in our argument, we show that a quasisymmetric parametrization of the Grushin plane by the Euclidean plane must also be geometrically quasiconformal. We also discuss some aspects of the Euclidean theory of quasiconformal maps, such as absolute continuity on almost every compact curve, not satisfied in the Grushin case.

8. Overconvergent Modular Forms and Perfectoid Shimura Curves

We give a new construction of overconvergent modular forms of arbitrary weights, defining them in terms of functions on certain affinoid subsets of Scholze's infinite-level modular curve. These affinoid subsets, and a certain canonical coordinate on them, play a role in our construction which is strongly analogous with the role of the upper half-plane and its coordinate `$z$' in the classical analytic theory of modular forms. As one application of these ideas, we define and study an overconvergent Eichler-Shimura map in the context of compact Shimura curves over $\mb{Q}$, proving stronger analogues of results of Andreatta-Iovita-Stevens.

Limit sets within curves where trajectories converge to

For continuously differentiable vector fields, we characterize the ω limit set of a trajectory converging to a compact curve Γ ⊂ R n . In particular, the limit set is either a fixed point or a continuum of fixed points if Γ is a simple open curve; otherwise, the limit set can in addition be either a closed orbit or a number of fixed points with compatibly oriented orbits connecting them. An implication of the result is a tightened-up version of the Poincaré–Bendixson theorem.

Ueda theory for compact curves with nodes

Let $C$ be a compact complex curve included in a non-singular complex surface such that the normal bundle is topologically trivial. Ueda studied complex analytic properties of a neighborhood of $C$ when $C$ is non-singular or is a rational curve with a node. We propose an analogue of Ueda's theory for the case where $C$ admits nodes. As an application, we study singular Hermitian metrics with semi-positive curvature on the anti-canonical bundle of the blow-up of the projective plane at nine points in arbitrary position.

On mother body measures with algebraic Cauchy transform

Below we discuss the existence of a mother body measure for the exterior inverse problem in potential theory in the complex plane. More exactly, we study the question of representability almost everywhere (a.e.) in \mathbb C of (a branch of) an irreducible algebraic function as the Cauchy transform of a signed measure supported on a finite number of compact semi-analytic curves and a finite number of isolated points. Firstly, we present a large class of algebraic functions for which there (conjecturally) always exists a positive measure with the above properties. This class was discovered in our earlier study of exactly solvable linear differential operators. Secondly, we investigate in detail the representability problem in the case when the Cauchy transform satisfies a quadratic equation with polynomial coefficients a.e. in \mathbb C . Several conjectures and open problems are posed.

A retrospective analysis of compact fluorescent lamp experience curves and their correlations to deployment programs

Experience curves are useful for understanding technology development and can aid in the design and analysis of market transformation programs. Here, we employ a novel approach to create experience curves, to examine both global and North American compact fluorescent lamp (CFL) data for the years 1990–2007. We move away from the prevailing method of fitting a single, constant, exponential curve to data and instead search for break points where changes in the learning rate may have occurred. Our analysis suggests a learning rate of approximately 21% for the period of 1990–1997, and 51% and 79% in global and North American datasets, respectively, after 1998. We use price data for this analysis; therefore our learning rates encompass developments beyond typical “learning by doing”, including supply chain impacts such as market competition. We examine correlations between North American learning rates and the initiation of new programs, abrupt technological advances, and economic and political events, and find an increased learning rate associated with design advancements and federal standards programs. Our findings support the use of segmented experience curves for retrospective and prospective technology analysis, and may imply that investments in technology programs have contributed to an increase of the CFL learning rate.

Fidelity vs. simplicity

Vector drawing is a popular representation in graphic design because of the precision, compactness and editability offered by parametric curves. However, prior work on line drawing vectorization focused solely on faithfully capturing input bitmaps, and largely overlooked the problem of producing a compact and editable curve network. As a result, existing algorithms tend to produce overly-complex drawings composed of many short curves and control points, especially in the presence of thick or sketchy lines that yield spurious curves at junctions. We propose the first vectorization algorithm that explicitly balances fidelity to the input bitmap with simplicity of the output, as measured by the number of curves and their degree. By casting this trade-off as a global optimization, our algorithm generates few yet accurate curves, and also disambiguates curve topology at junctions by favoring the simplest interpretations overall. We demonstrate the robustness of our algorithm on a variety of drawings, sketchy cartoons and rough design sketches.

Quadratic differentials (A(z − a)(z − b)/(z − c)2) dz 2 and algebraic Cauchy transform

We discuss the representability almost everywhere (a.e.) in C of an irreducible algebraic function as the Cauchy transform of a signed measure supported on a finite number of compact semi-analytic curves and a finite number of isolated points. This brings us to the study of trajectories of the particular family of quadratic differentials A(z − a)(z − b)×(z − c)−2 dz2. More precisely, we give a necessary and sufficient condition on the complex numbers a and b for these quadratic differentials to have finite critical trajectories. We also discuss all possible configurations of critical graphs.

On the $C^\alpha$-convergence of the Solution of the Chern-Ricci Flow on Elliptic Surfaces

We will study the Chern-Ricci flow on non-Kähler properly elliptic surfaces. These surfaces are compact complex surfaces whose first Betti number is odd, Kodaira dimension is equal to 1 and admit an elliptic fibration to a smooth compact curve. We will show that a solution of the Chern-Ricci flow converges in $C^\alpha$-topology on these elliptic surfaces by choosing a special initial metric.

The irreducible components of the moduli space of dihedral covers of algebraic curves

The main purpose of this paper is to introduce a new invariant for the action of a finite group G on a compact complex curve of genus g . With the aid of this invariant we achieve the classication of the components of the locus (in the moduli space) of curves admitting an effective action by the dihedral group D_n . This invariant has later been used in [11] where the results of Livingston [29] and of Dunfield and Thurston [17] have been extended to the ramified case.