The Poincaré Dual of a Geodesic Algebraic Curve in a Quotient of the 2-Ball

1981 ◽  
Vol 33 (2) ◽  
pp. 485-499 ◽  
Author(s):  
Stephen S. Kudla ◽  
John J. Millson
Keyword(s):  
Algebraic Curve ◽  
Inverse Image ◽  
Universal Covering ◽  
Explicit Construction ◽  
Holomorphic Curve ◽  
Volume Form ◽  
Harmonic Form ◽  
Bergman Metric ◽  
Kähler Structure ◽  
Totally Geodesic

We shall consider an irreducible, non-singular, totally geodesic holomorphic curve N in a compact quotient M = Γ\D of the unit ball D = {(z, w):|z|2 + |w|2 < 1} in C2 with the Kahler structure provided by the Bergman metric. The main result of this paper is an explicit construction of the harmonic form of type (1,1) which is dual to N. Our construction is as follows. Let p:D → Γ\D be the universal covering map. Choose a component D1 in the inverse image of N under p. The choice of D1 corresponds to choosing an embedding of the fundamental group of N into Γ. We denote the image by Γ1. Let π : D → D1 be the fiber bundle obtained by exponentiating the normal bundle of D1 in D. Let μ be the volume form of D1.

scholarly journals Almost Complex Surfaces in the Nearly Kähler SL(2,ℝ) × SL(2,ℝ)

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1160
Author(s):  
Elsa Ghandour ◽  
Luc Vrancken
Keyword(s):  
Complete Classification ◽  
Second Fundamental Form ◽  
Complex Surfaces ◽  
Kähler Structure ◽  
Totally Geodesic ◽  
Parallel Second Fundamental Form ◽  
Almost Complex ◽  
Nearly Kähler ◽  
Nearly Kähler Structure

The space S L ( 2 , R ) × S L ( 2 , R ) admits a natural homogeneous pseudo-Riemannian nearly Kähler structure. We investigate almost complex surfaces in this space. In particular, we obtain a complete classification of the totally geodesic almost complex surfaces and of the almost complex surfaces with parallel second fundamental form.

scholarly journals Finitely curved orbits of complex polynomial vector fields

2007 ◽  
Vol 79 (1) ◽  
pp. 13-16
Author(s):  
Albetã C. Mafra
Keyword(s):  
Vector Field ◽  
Gaussian Curvature ◽  
Algebraic Curve ◽  
Vector Fields ◽  
Complex Polynomial ◽  
Holomorphic Foliations ◽  
Holomorphic Curve ◽  
Polynomial Vector ◽  
Polynomial Vector Fields

This note is about the geometry of holomorphic foliations. Let X be a polynomial vector field with isolated singularities on C². We announce some results regarding two problems: 1. Given a finitely curved orbit L of X, under which conditions is L algebraic? 2. If X has some non-algebraic finitely curved orbit L what is the classification of X? Problem 1 is related to the following question: Let C <FONT FACE=Symbol>Ì</FONT> C² be a holomorphic curve which has finite total Gaussian curvature. IsC contained in an algebraic curve?

scholarly journals EXTENSION OF FUNCTORS FOR ALGEBRAS OF FORMAL DEFORMATION

2013 ◽  
Vol 56 (1) ◽  
pp. 103-141
Author(s):  
ANA RITA MARTINS ◽  
TERESA MONTEIRO FERNANDES ◽  
DAVID RAIMUNDO
Keyword(s):  
Fourier Transform ◽  
Open Subset ◽  
Inverse Image ◽  
Explicit Construction ◽  
Comparison Theorems ◽  
Complex Manifolds ◽  
Formal Parameter ◽  
Formal Deformation ◽  
Characteristic Case ◽  
Vanishing Cycles

AbstractSuppose we are given complex manifoldsXandYtogether with substacks$\mathcal{S}$and$\mathcal{S}'$of modules over algebras of formal deformation$\mathcal{A}$onXand$\mathcal{A}'$onY, respectively. Also, suppose we are given a functor Φ from the category of open subsets ofXto the category of open subsets ofYtogether with a functorFof prestacks from$\mathcal{S}$to$\mathcal{S}'\circ\Phi$. Then we give conditions for the existence of a canonical functor, extension ofFto the category of coherent$\mathcal{A}$-modules such that the cohomology associated to the action of the formal parameter$\hbar$takes values in$\mathcal{S}$. We give an explicit construction and prove that when the initial functorFis exact on each open subset, so is its extension. Our construction permits to extend the functors of inverse image, Fourier transform, specialisation and micro-localisation, nearby and vanishing cycles in the framework of$\mathcal{D}[[\hbar]]$-modules. We also obtain the Cauchy–Kowalewskaia–Kashiwara theorem in the non-characteristic case as well as comparison theorems for regular holonomic$\mathcal{D}[[\hbar]]$-modules and a coherency criterion for proper direct images of good$\mathcal{D}[[\hbar]]$-modules.

Bergman kernel on Riemann surfaces and Kähler metric on symmetric products

2019 ◽  
Vol 30 (14) ◽  
pp. 1950071
Author(s):  
Anilatmaja Aryasomayajula ◽  
Indranil Biswas
Keyword(s):  
Bergman Kernel ◽  
Riemann Surfaces ◽  
Symmetric Product ◽  
Volume Form ◽  
Bergman Metric ◽  
Poincaré Metric ◽  
Kähler Metric ◽  
Fixed Dimension ◽  
Holomorphic Sections ◽  
Kahler Metric

Let [Formula: see text] be a compact hyperbolic Riemann surface equipped with the Poincaré metric. For any integer [Formula: see text], we investigate the Bergman kernel associated to the holomorphic Hermitian line bundle [Formula: see text], where [Formula: see text] is the holomorphic cotangent bundle of [Formula: see text]. Our first main result estimates the corresponding Bergman metric on [Formula: see text] in terms of the Poincaré metric. We then consider a certain natural embedding of the symmetric product of [Formula: see text] into a Grassmannian parametrizing subspaces of fixed dimension of the space of all global holomorphic sections of [Formula: see text]. The Fubini–Study metric on the Grassmannian restricts to a Kähler metric on the symmetric product of [Formula: see text]. The volume form for this restricted metric on the symmetric product is estimated in terms of the Bergman kernel of [Formula: see text] and the volume form for the orbifold Kähler form on the symmetric product given by the Poincaré metric on [Formula: see text].

scholarly journals On the Calabi-Yau equation in the Kodaira-Thurston manifold

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Luigi Vezzoni
Keyword(s):  
Higher Dimensions ◽  
Invariant Function ◽  
Complex Structures ◽  
Volume Form ◽  
Kähler Structure ◽  
Almost Complex Structures ◽  
Almost Complex ◽  
Almost Kähler

AbstractWe review some previous results about the Calabi-Yau equation on the Kodaira-Thurston manifold equipped with an invariant almost-Kähler structure and assuming the volume form T2-invariant. In particular, we observe that under some restrictions the problem is reduced to aMonge-Ampère equation by using the ansatz ˜~ω = Ω− dJdu + da, where u is a T2-invariant function and a is a 1-form depending on u. Furthermore, we extend our analysis to non-invariant almost-complex structures by considering some basic cases and we finally take into account a generalization to higher dimensions.

scholarly journals On area stationary surfaces in the space of oriented geodesics of hyperbolic 3-space

2012 ◽  
Vol 111 (2) ◽  
pp. 187
Author(s):  
Nikos Georgiou
Keyword(s):  
Parameter Family ◽  
Holomorphic Curve ◽  
Kähler Structure ◽  
Stationary Surface ◽  
The Family ◽  
Rotationally Symmetric ◽  
Parallel Surfaces ◽  
Space Of Oriented Geodesics ◽  
Two Parameter

We study area-stationary surfaces in the space $\mathbf{L}(\mathbf{H}^3)$ of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. We prove that every holomorphic curve in $\mathbf{L}(\mathbf{H}^3)$ is an area-stationary surface. We then classify Lagrangian area-stationary surfaces $\Sigma$ in $\mathbf{L}(\mathbf{H}^3)$ and prove that the family of parallel surfaces in $\mathbf{H}^3$ orthogonal to the geodesics $\gamma\in \Sigma$ form a family of equidistant tubes around a geodesic. Finally we find an example of a two parameter family of rotationally symmetric area-stationary surfaces that are neither Lagrangian nor holomorphic.

Dosimetry and optimization in digital radiography based on the contrast-detail resolution

2018 ◽  
Vol 6 (3) ◽  
Author(s):  
Wilson Otto Gomes Batista ◽  
Alexandre Gomes De Carvalho
Keyword(s):  
Image Quality ◽  
Inverse Image ◽  
Point Of View ◽  
Complete Analysis ◽  
Portable Devices ◽  
Evaluation Tool ◽  
Air Kerma ◽  
Direct Radiography ◽  
Definition Of ◽  
Entrance Surface Air Kerma

Contrast-detail (C-D) curves are useful in evaluating the radiographic image quality in a global way. The objective of the present study was to obtain the C-D curves and the inverse image quality figure. Both of these parameters were used as an evaluation tool for abdominal and chest imaging protocols. The C-D curves were obtained with the phantom CDRAD 2.0 in computerized radiography and the direct radiography systems (including portable devices). The protocols were 90 and 102 kV in the range of 2 to 20 mAs for the chest and 80 kV in the range of 10 to 80 mAs for the abdomen. The incident air kerma values were evaluated with a solid state sensor. The analysis of these C-D curves help to identify which technique would allow a lower value of the entrance surface air kerma, Ke, while maintaining the image quality from the point of view of C-D detectability. The results showed that the inverse image quality figure, IQFinv, varied little throughout the range of mAs, while the value of Ke varied linearly directly with the mAs values. Also, the complete analysis of the curves indicated that there was an increase in the definition of the details with increasing mAs. It can be concluded that, in the transition phase for the use of the new receptors, it is necessary to evaluate and adjust the practised protocols to ensure, at a minimum, the same levels of the image quality, taking into account the aspects of the radiation protection of the patient.

A note on invariant submanifolds of CR-integrable almost Kenmotsu manifolds

Filomat ◽  
2017 ◽  
Vol 31 (19) ◽  
pp. 6211-6218 ◽  
Author(s):  
Young Suh ◽  
Krishanu Mandal ◽  
Uday De
Keyword(s):  
Invariant Submanifold ◽  
Kenmotsu Manifold ◽  
Totally Geodesic ◽  
Almost Kenmotsu Manifold ◽  
Kenmotsu Manifolds ◽  
Almost Kenmotsu Manifolds ◽  
Invariant Submanifolds

The present paper deals with invariant submanifolds of CR-integrable almost Kenmotsu manifolds. Among others it is proved that every invariant submanifold of a CR-integrable (k,?)'-almost Kenmotsu manifold with k < -1 is totally geodesic. Finally, we construct an example of an invariant submanifold of a CR-integrable (k,?)'-almost Kenmotsu manifold which is totally geodesic.

scholarly journals Discrete Nonlinear Schrödinger Systems for Periodic Media with Nonlocal Nonlinearity: The Case of Nematic Liquid Crystals

2021 ◽  
Vol 11 (10) ◽  
pp. 4420
Author(s):  
Panayotis Panayotaros
Keyword(s):  
Differential Equation ◽  
Infinite System ◽  
Linear Part ◽  
Optical Power ◽  
Explicit Construction ◽  
Translation Invariance ◽  
Nonlocal Nonlinearity ◽  
Wannier Functions ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrödinger Systems

We study properties of an infinite system of discrete nonlinear Schrödinger equations that is equivalent to a coupled Schrödinger-elliptic differential equation with periodic coefficients. The differential equation was derived as a model for laser beam propagation in optical waveguide arrays in a nematic liquid crystal substrate and can be relevant to related systems with nonlocal nonlinearities. The infinite system is obtained by expanding the relevant physical quantities in a Wannier function basis associated to a periodic Schrödinger operator appearing in the problem. We show that the model can describe stable beams, and we estimate the optical power at different length scales. The main result of the paper is the Hamiltonian structure of the infinite system, assuming that the Wannier functions are real. We also give an explicit construction of real Wannier functions, and examine translation invariance properties of the linear part of the system in the Wannier basis.

scholarly journals Singular BPS boundary conditions in $$ \mathcal{N} $$ = (2, 2) supersymmetric gauge theories

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Tadashi Okazaki ◽  
Douglas J. Smith
Keyword(s):  
String Theory ◽  
Boundary Conditions ◽  
Gauge Theories ◽  
Holomorphic Curve ◽  
Two Dimensional ◽  
Supersymmetric Gauge Theories ◽  
Special Lagrangian ◽  
Supersymmetric Gauge ◽  
M Theory

Abstract We derive general BPS boundary conditions in two-dimensional $$ \mathcal{N} $$ N = (2, 2) supersymmetric gauge theories. We analyze the solutions of these boundary conditions, and in particular those that allow the bulk fields to have poles at the boundary. We also present the brane configurations for the half- and quarter-BPS boundary conditions of the $$ \mathcal{N} $$ N = (2, 2) supersymmetric gauge theories in terms of branes in Type IIA string theory. We find that both A-type and B-type brane configurations are lifted to M-theory as a system of M2-branes ending on an M5-brane wrapped on a product of a holomorphic curve in ℂ2 with a special Lagrangian 3-cycle in ℂ3.

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