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Journal articles on the topic 'Holomorphic quadratic differentials'

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Published: 6 September 2023

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1

Große, Nadine, and Melanie Rupflin. "Holomorphic quadratic differentials dual to Fenchel–Nielsen coordinates." Annals of Global Analysis and Geometry 55, no. 3 (November 16, 2018): 479–507. http://dx.doi.org/10.1007/s10455-018-9636-y.

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2

Kenyon, Richard, and Wai Yeung Lam. "Holomorphic quadratic differentials on graphs and the chromatic polynomial." Journal of Combinatorial Theory, Series A 170 (February 2020): 105140. http://dx.doi.org/10.1016/j.jcta.2019.105140.

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3

AU, THOMAS KWOK-KEUNG, and TOM YAU-HENG WAN. "PRESCRIBED HORIZONTAL AND VERTICAL TREES PROBLEM OF QUADRATIC DIFFERENTIALS." Communications in Contemporary Mathematics 08, no. 03 (June 2006): 381–99. http://dx.doi.org/10.1142/s0219199706002155.

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A sufficient condition for the existence of holomorphic quadratic differential on a non-compact simply-connected Riemann surface with prescribed horizontal and vertical trees is obtained. In particular, for any pair of complete ℝ-trees of finite vertices with (n + 2) infinite edges, there exists a polynomial quadratic differential on ℂ of degree n such that the associated vertical and horizontal trees are isometric to the given pair.

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4

Yao, Guowu. "A note on holomorphic quadratic differentials on the unit disk." Kodai Mathematical Journal 39, no. 1 (March 2016): 72–79. http://dx.doi.org/10.2996/kmj/1458651692.

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5

Fernández, Isabel, and Pablo Mira. "Holomorphic quadratic differentials and the Bernstein problem in Heisenberg space." Transactions of the American Mathematical Society 361, no. 11 (June 22, 2009): 5737–52. http://dx.doi.org/10.1090/s0002-9947-09-04645-5.

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6

Chang, Zhe. "The Holomorphic Quadratic Differentials of Amplitudes for Strings with Boundaries." Communications in Theoretical Physics 13, no. 1 (January 1990): 49–56. http://dx.doi.org/10.1088/0253-6102/13/1/49.

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7

KIRSCHNER, R. "BOUNDARY REPARAMETRIZATIONS AS ADDITIONAL MODULI FOR THE STRING PROPAGATOR." Modern Physics Letters A 04, no. 03 (February 1989): 283–91. http://dx.doi.org/10.1142/s0217732389000356.

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Analyzing the Polyakov integral on surfaces with boundaries, where the values of the string variables are fixed, we use the observation that there are more holomorphic quadratic differentials besides those obtained as restrictions from the Schottky double. They are naturally related to boundary reparametrizations. The corresponding additional moduli are used to express the integration over metrices. Some details are given for the vacuum functional and the propagator.

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8

Matone, Marco, and Roberto Volpato. "Linear relations among holomorphic quadratic differentials and induced Siegel's metric on Mg." Journal of Mathematical Physics 52, no. 10 (October 2011): 102305. http://dx.doi.org/10.1063/1.3653550.

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9

Sugawa, Toshiyuki. "A conformally invariant metric on Riemann surfaces associated with integrable holomorphic quadratic differentials." Mathematische Zeitschrift 266, no. 3 (August 11, 2009): 645–64. http://dx.doi.org/10.1007/s00209-009-0590-z.

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10

ZHANG, C. "SINGULARITIES OF QUADRATIC DIFFERENTIALS AND EXTREMAL TEICHMÜLLER MAPPINGS DEFINED BY DEHN TWISTS." Journal of the Australian Mathematical Society 87, no. 2 (October 2009): 275–88. http://dx.doi.org/10.1017/s1446788709000032.

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AbstractLet S be a Riemann surface of finite type. Let ω be a pseudo-Anosov map of S that is obtained from Dehn twists along two families {A,B} of simple closed geodesics that fill S. Then ω can be realized as an extremal Teichmüller mapping on a surface of the same type (also denoted by S). Let ϕ be the corresponding holomorphic quadratic differential on S. We show that under certain conditions all possible nonpuncture zeros of ϕ stay away from all closures of once punctured disk components of S∖{A,B}, and the closure of each disk component of S∖{A,B} contains at most one zero of ϕ. As a consequence, we show that the number of distinct zeros and poles of ϕ is less than or equal to the number of components of S∖{A,B}.

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11

Huo, Sheng Jin. "A note on the density of a subset of all integrable holomorphic quadratic differentials." Acta Mathematica Sinica, English Series 30, no. 5 (April 15, 2014): 793–96. http://dx.doi.org/10.1007/s10114-014-3224-2.

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12

Earle, Clifford J., and V. Markovic. "Isometries between the spaces of $L^1$ holomorphic quadratic differentials on Riemann surfaces of finite type." Duke Mathematical Journal 120, no. 2 (November 2003): 433–40. http://dx.doi.org/10.1215/s0012-7094-03-12029-3.

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13

DIAS, KEALEY. "Enumerating combinatorial classes of the complex polynomial vector fields in ℂ." Ergodic Theory and Dynamical Systems 33, no. 2 (February 20, 2012): 416–40. http://dx.doi.org/10.1017/s0143385711000952.

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AbstractIn order to understand the parameter space Ξd of monic and centered complex polynomial vector fields in ℂ of degree d, decomposed by the combinatorial classes of the vector fields, it is interesting to know the number of loci in parameter space consisting of vector fields with the same combinatorial data (corresponding to topological classification with fixed separatrices at infinity). This paper answers questions posed by Adam L. Epstein and Tan Lei about the total number of combinatorial classes and the number of combinatorial classes corresponding to loci of a specific (real) dimension q in parameter space, for fixed degree d; these numbers are denoted by cd and cd,q, respectively. These results are extensions of a result by Douady, Estrada, and Sentenac, which shows that the number of combinatorial classes of the structurally stable complex polynomial vector fields in ℂ of degree d is the Catalan number Cd−1. We show that enumerating the combinatorial classes is equivalent to a so-called bracketing problem. Then we analyze the generating functions and find closed-form expressions for cd and cd,q, and we furthermore make an asymptotic analysis of these sequences for d tending to ∞. These results are also applicable to special classes of quadratic and Abelian differentials and singular holomorphic foliations of the plane.

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14

Alvarez-Parrilla, Alvaro, and Jesús Muciño-Raymundo. "Symmetries of complex analytic vector fields with an essential singularity on the Riemann sphere." Advances in Geometry 21, no. 4 (October 1, 2021): 483–504. http://dx.doi.org/10.1515/advgeom-2021-0002.

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Abstract We consider the family ℰ (s, r, d) of all singular complex analytic vector fields X ( z ) = Q ( z ) P ( z ) e E ( z ) ∂ ∂ z $X(z)=\frac{Q(z)}{P(z)}{{e}^{E(z)}}\frac{\partial }{\partial z}$ on the Riemann sphere where Q, P, ℰ are polynomials with deg Q = s, deg P = r and deg ℰ = d ≥ 1. Using the pullback action of the affine group Aut(ℂ) and the divisors for X, we calculate the isotropy groups Aut(ℂ) X of discrete symmetries for X ∈ ℰ (s, r, d). The subfamily ℰ (s, r, d)id of those X with trivial isotropy group in Aut(ℂ) is endowed with a holomorphic trivial principal Aut(ℂ)-bundle structure. A necessary and sufficient arithmetic condition on s, r, d ensuring the equality ℰ (s, r, d) = ℰ (s, r, d)id is presented. Moreover, those X ∈ ℰ (s, r, d) \ ℰ (s, r, d)id with non-trivial isotropy are realized. This yields explicit global normal forms for all X ∈ ℰ (s, r, d). A natural dictionary between analytic tensors, vector fields, 1-forms, orientable quadratic differentials and functions on Riemann surfaces M is extended as follows. In the presence of nontrivial discrete symmetries Γ < Aut(M), the dictionary describes the correspondence between Γ-invariant tensors on M and tensors on M /Γ.

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15

Alvarez-Parrilla, Alvaro, and Jesús Muciño-Raymundo. "Symmetries of complex analytic vector fields with an essential singularity on the Riemann sphere." Advances in Geometry 21, no. 4 (October 1, 2021): 483–504. http://dx.doi.org/10.1515/advgeom-2021-0002.

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Abstract We consider the family ℰ (s, r, d) of all singular complex analytic vector fields X ( z ) = Q ( z ) P ( z ) e E ( z ) ∂ ∂ z $X(z)=\frac{Q(z)}{P(z)}{{e}^{E(z)}}\frac{\partial }{\partial z}$ on the Riemann sphere where Q, P, ℰ are polynomials with deg Q = s, deg P = r and deg ℰ = d ≥ 1. Using the pullback action of the affine group Aut(ℂ) and the divisors for X, we calculate the isotropy groups Aut(ℂ) X of discrete symmetries for X ∈ ℰ (s, r, d). The subfamily ℰ (s, r, d)id of those X with trivial isotropy group in Aut(ℂ) is endowed with a holomorphic trivial principal Aut(ℂ)-bundle structure. A necessary and sufficient arithmetic condition on s, r, d ensuring the equality ℰ (s, r, d) = ℰ (s, r, d)id is presented. Moreover, those X ∈ ℰ (s, r, d) \ ℰ (s, r, d)id with non-trivial isotropy are realized. This yields explicit global normal forms for all X ∈ ℰ (s, r, d). A natural dictionary between analytic tensors, vector fields, 1-forms, orientable quadratic differentials and functions on Riemann surfaces M is extended as follows. In the presence of nontrivial discrete symmetries Γ < Aut(M), the dictionary describes the correspondence between Γ-invariant tensors on M and tensors on M /Γ.

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16

Masur, Howard. "The growth rate of trajectories of a quadratic differential." Ergodic Theory and Dynamical Systems 10, no. 1 (March 1990): 151–76. http://dx.doi.org/10.1017/s0143385700005459.

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AbstractSupposeqis a holomorphic quadratic differential on a compact Riemann surface of genusg≥ 2. Thenqdefines a metric, flat except at the zeroes. A saddle connection is a geodesic joining two zeroes with no zeroes in its interior. This paper shows the asymptotic growth rate of the number of saddles of length at mostTis at most quadratic inT. An application is given to billiards.

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17

Nishikawa, Seiki. "Harmonic maps from Riemann surfaces into complex Finsler manifolds." International Journal of Mathematics 26, no. 06 (June 2015): 1541010. http://dx.doi.org/10.1142/s0129167x15410104.

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Given a smooth map from a compact Riemann surface to a complex manifold equipped with a strongly pseudoconvex complex Finsler metric, we define the [Formula: see text]-energy of the map, whose absolute minimum is attained by a holomorphic map. A harmonic map is then defined to be a stationary map of the [Formula: see text]-energy functional. We prove that with each harmonic map is associated a holomorphic quadratic differential on the domain, which vanishes if the map is weakly conformal. Also, under the condition that the metric be weakly Kähler, we determine the second variation of the functional, and prove that any [Formula: see text]-energy minimizing harmonic map from the Riemann sphere to a weakly Kähler Finsler manifold of positive curvature is either holomorphic or anti-holomorphic.

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18

Cheng, Tao. "Decomposition of integrable holomorphic quadratic differential on Riemann surface of infinite type." Science China Mathematics 53, no. 8 (July 22, 2010): 2039–44. http://dx.doi.org/10.1007/s11425-010-4055-y.

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19

Martin, Gaven J., and Cong Yao. "The Teichmüller problem for L^p-means of distortion." Annales Fennici Mathematici 47, no. 2 (August 30, 2022): 1099–108. http://dx.doi.org/10.54330/afm.121468.

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Teichmüller's problem from 1944 is this: Given \(x\in [0,1)\) find and describe the extremal quasiconformal map \(f\colon\mathbb{D}\to\mathbb{D}\), \(f|\partial \mathbb{D}=\) identity and \(f(0)=-x\leq 0\). We consider this problem in the setting of minimisers of \(L^p\)-mean distortion. The classical result is that there is an extremal map of Teichmüller type with associated holomorphic quadratic differential having a pole of order one at 0, if \(x\neq 0\). For the \(L^1\)-norm it is known that there can be no locally quasiconformal minimiser unless \(x=0\). Here we show that for \(1\leq p<\infty\) there is a minimiser in a weak class and an associated Ahlfors-Hopf holomorphic quadratic differential with a pole of order 1 at \(f(0)=-x\). However, this minimiser cannot be in \(W^{1,2}_{loc}(\mathbb{D})\) unless \(x=0\) and \(f=\) identity. Hence no minimiser for the \(L^p\)-Teichmüller problem can be locally quasiconformal other than the identity. Similar statements holds for minimisers of the exponential norm of distortion. We also use our earlier work to show that as \(p\to\infty\), the weak \(L^p\)-minimisers converge locally uniformly in \(\mathbb{D}\) to the extremal quasiconformal Teichmüller mapping, and that as \(p\to 1\) the weak \(L^p\)-minimisers converge locally uniformly in \(\mathbb{D}\) to the identity.

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20

Shi, Qingtian. "Equivalent characterizations of harmonic Teichmüller mappings." AIMS Mathematics 7, no. 6 (2022): 11015–23. http://dx.doi.org/10.3934/math.2022615.

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<abstract><p>In this paper, three equivalent conditions of $ \rho $-harmonic Teichmüller mapping are given firstly. As an application, we investigate the relationship between a $ \rho $-harmonic Teichmüller mapping and its associated holomorphic quadratic differential and obtain a relatively simple method to prove Theorem 2.1 in <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>. Furthermore, the representation theorem of $ 1/|\omega|^{2} $-harmonic Teichmüller mappings is given as a by-product. Our results extend the corresponding researches of harmonic Teichmüller mappings.</p></abstract>

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21

VEECH, WILLIAM A. "Measures supported on the set of uniquely ergodic directions of an arbitrary holomorphic 1-form." Ergodic Theory and Dynamical Systems 19, no. 4 (August 1999): 1093–109. http://dx.doi.org/10.1017/s014338579913390x.

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We introduce a set, $Q({\bf T})$, of Borel probability measures on the circle such that each $\mu\in Q({\bf T})$ obeys the conclusion of the Kerckhoff–Masur–Smillie theorem [3]: if $q$ is a meromorphic quadratic differential with at worst simple poles on a closed Riemann surface, then for each $\mu\in Q({\bf T})$ and $\mu$-a.e. $\zeta\in{\bf T}$, $\zeta q$ has uniquely ergodic vertical foliation. As an example, the normalized Cantor–Lebesgue measure belongs to $Q({\bf T})$. The analysis also yields an analogue, for the Teichmüller horocycle flow, of a theorem of Dani: every locally finite ergodic invariant measure for the Teichmüller horocycle flow is finite.

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22

Richter, Wolf-Dieter. "On Hyperbolic Complex Numbers." Applied Sciences 12, no. 12 (June 8, 2022): 5844. http://dx.doi.org/10.3390/app12125844.

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For dimensions two, three and four, we derive hyperbolic complex algebraic structures on the basis of suitably defined vector products and powers which allow in a standard way a series definitions of the hyperbolic vector exponential function. In doing so, we both modify arrow multiplication, which, according to Feynman, is fundamental for quantum electrodynamics, and we give a geometric explanation of why in a certain situation it is natural to define random vector products. Through the interplay of vector algebra, geometry and complex analysis, we extend a systematic approach previously developed for various other complex algebraic structures to the field of hyperbolic complex numbers. We discuss a quadratic vector equation and the property of hyperbolically holomorphic functions of satisfying hyperbolically modified Cauchy–Riemann differential equations and also give a proof of an Euler type formula based on series expansion.

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23

Jakobsen, Hans Plesner. "Algebras of variable coefficient quantized differential operators." Journal of Mathematical Physics 63, no. 8 (August 1, 2022): 081704. http://dx.doi.org/10.1063/5.0091631.

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In the framework of (vector valued) quantized holomorphic functions defined on non-commutative spaces, “quantized Hermitian symmetric spaces,” an obvious problem is to describe (quantum) holomorphically induced representations in terms of some manageable structures. An intimately related problem is to decide what the algebras of quantized differential operators with variable coefficients should be. It is an immediate point that even zeroth order operators, given as multiplications by polynomials, have to be specified as, e.g., left or right multiplication operators since the polynomial algebras are replaced by quadratic, non-commutative algebras. In the settings we are interested in, there are bilinear pairings that allow us to define differential operators as duals of multiplication operators. Indeed, there are different choices of pairings which lead to quite different results. We consider three different pairings while specializing to [Formula: see text]. The pairings are between quantized generalized Verma modules and quantized holomorphically induced modules. It is a natural demand that the corresponding representations can be expressed by (matrix valued) differential operators. We show that a quantum Weyl algebra [Formula: see text] introduced by Hayashi [Commun. Math. Phys. 127(1), 129–144 (1990)] plays a fundamental role. In fact, for one pairing, the algebra of differential operators, though inherently depending on a choice of basis, is precisely matrices over [Formula: see text]. We determine explicitly the form of the (quantum) holomorphically induced representations and determine, for the different pairings, if they can be expressed by differential operators.

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24

Maslikov, Alexander, and Guennady Volkov. "Ternary SU(3)-group symmetry and its possible applications in hadron-quark substructure. Towards a new spinor-fermion structure." EPJ Web of Conferences 204 (2019): 02007. http://dx.doi.org/10.1051/epjconf/201920402007.

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The questions on the existence of the three color quark symmetry and three quark-lepton generations could have the origin associated with the new exotic symmetries outside the Cartan-Killing-Lie algebras/groups. Our long-term search for these symmetries has been began with our Calabi-Yau space classification on the basis of the n-ary algebra for the reflexive projective numbers and led us to the expansion of the binary n = 2 complex and hyper complex numbers in the framework of the n-ary complex and hyper-complex numbers with n = 3, 4, … where we constructed new Abelian and non-Abelian symmetries. We have studied then norm-division properties of the Abelian nary complex numbers and have built the infinite chain of the Abelian groups U(n–1) = [U(1) × … × U(1)](n–1). We have developed the n-ary holomorphic (polymorphic) analysis on the n-ary complex space NC{n}, which led us to the generalization of the quadratic Laplace equations for the harmonic functions. The generalized Laplace equations for the n-ary harmonic functions give us the n-th order homogeneous differential equations which are invariant with respect to the Abelian n-ary groups U(n–1) and with some new spatial properties. Further consideration of the non-Abelian n-ary hyper-complex numbers opens the infinite series of the non-Abelian TnSU(n)-Lie groups(n=3,4,…) and its corresponding tnsu(n) algebras. One of the exceptional features of these symmetry groups is the appearance of some new n-dimensional spinors that could lead to an extension of the concept of the SU(2)-spin, to the appearance of n-dimensional quantum structures -exotic “n-spinor” matter(n = 3, 4, … - maarcrions). It is natural to assume that these new exotic “quantum spinor states” could be candidates for the pra-matter of the quark-charge leptons or/and for the dark matter. We will be also interested in the detection of the exotic quantum ’n-spinor” matter in the neutrino and hadron experiments.

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25

Korotkin, Dmitry, and Peter Zograf. "Tau Function and Moduli of Meromorphic Quadratic Differential." Symmetry, Integrability and Geometry: Methods and Applications, January 3, 2022. http://dx.doi.org/10.3842/sigma.2022.001.

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The Bergman tau functions are applied to the study of the Picard group of moduli spaces of quadratic differentials with at most n simple poles on genus g complex algebraic curves. This generalizes our previous results on moduli spaces of holomorphic quadratic differentials.

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26

Gupta, Subhojoy. "Harmonic maps and wild Teichmüller spaces." Journal of Topology and Analysis, May 6, 2019, 1–45. http://dx.doi.org/10.1142/s1793525320500156.

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We use meromorphic quadratic differentials with higher order poles to parametrize the Teichmüller space of crowned hyperbolic surfaces. Such a surface is obtained on uniformizing a compact Riemann surface with marked points on its boundary components, and has noncompact ends with boundary cusps. This extends Wolf’s parametrization of the Teichmüller space of a closed surface using holomorphic quadratic differentials. Our proof involves showing the existence of a harmonic map from a punctured Riemann surface to a crowned hyperbolic surface, with prescribed principal parts of its Hopf differential which determine the geometry of the map near the punctures.

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27

Zhao, Hui, Shaodong Wang, and Wencheng Wang. "Global Conformal Parameterization via an Implementation of Holomorphic Quadratic Differentials." IEEE Transactions on Visualization and Computer Graphics, 2020, 1. http://dx.doi.org/10.1109/tvcg.2020.3016574.

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28

Brown, Francis, and Clément Dupont. "Single-valued integration and double copy." Journal für die reine und angewandte Mathematik (Crelles Journal), December 15, 2020. http://dx.doi.org/10.1515/crelle-2020-0042.

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AbstractIn this paper, we study a single-valued integration pairing between differential forms and dual differential forms which subsumes some classical constructions in mathematics and physics. It can be interpreted as a p-adic period pairing at the infinite prime. The single-valued integration pairing is defined by transporting the action of complex conjugation from singular to de Rham cohomology via the comparison isomorphism. We show how quite general families of period integrals admit canonical single-valued versions and prove some general formulae for them. This implies an elementary “double copy” formula expressing certain singular volume integrals over the complex points of a smooth projective variety as a quadratic expression in ordinary period integrals of half the dimension. We provide several examples, including non-holomorphic modular forms, archimedean Néron–Tate heights on curves, single-valued multiple zeta values and polylogarithms. The results of the present paper are used in [F. Brown and C. Dupont, Single-valued integration and superstring amplitudes in genus zero, preprint 2019, https://arxiv.org/abs/1910.01107] to prove a recent conjecture of Stieberger which relates the coefficients in a Laurent expansion of two different kinds of periods of twisted cohomology on the moduli spaces of curves {\mathcal{M}_{0,n}} of genus zero with n marked points. We also study a morphism between certain rings of “motivic” periods, called the de Rham projection, which provides a bridge between complex periods and single-valued periods in many situations of interest.

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