Consider a logharmonic polynomial; that is, a product of the form $p(z)\overline{q(z)}$, where $p$, $q$ are holomorphic polynomials. Assume $q$ is linear and denote by $n$ the degree of $p$. It was recently shown in arXiv:2302.04339 [math.CV] that the valence of such a logharmonic polynomial is at most $3n-1$; in this paper we show that their $3n-1$ upper bound is sharp. Together with the work of arXiv:2302.04339 [math.CV], this resolves a conjecture of Bshouty and Hengartner.
Genus one amplitude for topological strings on Calabi–Yau 3-folds can be computed using mirror symmetry: The partition function at genus one gets mapped to a holomorphic version of Ray–Singer torsion on the mirror Calabi–Yau. On the other hand it can be shown by a physical argument that this gives a curvature squared correction term to the gravitational action. This in paticular leads to an effective quantum gravity cutoff known as the species scale, which varies over moduli space of Calabi–Yau manifolds. This resolves some of the puzzles associated to the entropy of small black holes when there are a large number of light species of particles. Thus Ray–Singer torsion, via its connection to topological strings at genus one, provides a measure of light degrees of freedom of four dimensional N = 2 \mathcal {N}=2 supergravity theories.
This paper is a continuation of authors work: Fatou and Julia like sets, Ukranian Math. J., to appear/arXiv:2006.08308[math.CV](see [5]). Here, we introduce escaping like set and generalized escaping like set for a family of holomorphic functions on an arbitrary domain, and establish some distinctive properties of these sets. The connectedness of the Julia like set is also proved.
ABSTRACT We get sharp pointwise estimates for the gradient of Pf, where P is Bergman projection in terms of -norm of function f defined in . Using limiting argument we transfer this result to Cauchy projection on and hence, the optimal gradient estimates of solution of -problem, thus extending results from Kalaj, Vujadinović [Norm of the Bergman projection onto the Bloch space. J Oper Theory. 2015;73(1):113–126], Kalaj, Marković [Optimal estimates for the gradient of harmonic functions in the unit disk. Complex Anal Oper Theory. 2013;7:1167–1183], Melentijević [Norm of the Bergman projection onto the Bloch space with -invariant gradient norm. arXiv 1711.08719[math.CV]]. As corollaries we get the sharp gradient estimate of a function in Hardy and Bergman spaces and exact norms of Cauchy projection acting into the Bloch space equipped with several (quasi)-norms.
In the present paper, it is found conditions on the complex coefficient of the Beltrami equations with the degeneration of the uniform ellipticity in the unit disk under which their generalized homeomorphic solutions are continuous by Hölder on the boundary. These results can be applied to the investigations of various boundary value problems for the Beltrami equations. In a series of recent papers, under the study of the boundary value problems of Dirichlet, Hilbert, Neumann, Poincare and Riemann with arbitrary measurable boundary data for the Beltrami equations as well as for the generalizations of the Laplace equation in anisotropic and inhomogeneous media, it was applied the logarithmic capacity, see e.g. Gutlyanskii V., Ryazanov V., Yefimushkin A. On the boundary value problems for quasiconformal functions in the plane // Ukr. Mat. Visn. - 2015. - 12, no. 3. - P. 363-389; transl. in J. Math. Sci. (N.Y.) - 2016. - 214, no. 2. - P. 200-219; Gutlyanskii V., Ryazanov V., Yefimushkin A. On a new approach to the study of plane boundary-value problems // Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki. - 2017. - No. 4. - P. 12-18; Yefimushkin A. On Neumann and Poincare Problems in A-harmonic Analysis // Advances in Analysis. - 2016. - 1, no. 2. - P. 114-120; Efimushkin A., Ryazanov V. On the Riemann-Hilbert problem for the Beltrami equations in quasidisks // Ukr. Mat. Visn. - 2015. - 12, no. 2. - P. 190–209; transl. in J. Math. Sci. (N.Y.) - 2015. - 211, no. 5. - P. 646–659; Yefimushkin A., Ryazanov V. On the Riemann–Hilbert Problem for the Beltrami Equations // Contemp. Math. - 2016. - 667. - P. 299-316; Gutlyanskii V., Ryazanov V., Yakubov E., Yefimushkin A. On Hilbert problem for Beltrami equation in quasihyperbolic domains // ArXiv.org: 1807.09578v3 [math.CV] 1 Nov 2018, 28 pp. As well known, the logarithmic capacity of a set coincides with the so-called transfinite diameter of the set. This geometric characteristic implies that sets of logarithmic capacity zero and, as a consequence, measurable functions with respect to logarithmic capacity are invariant under mappings that are continuous by Hölder. That circumstance is a motivation of our research. Let \(D\) be a domain in the complex plane \(\mathbb C\) and let \(\mu: D\to\mathbb C\) be a measurable function with \( |\mu(z)| \lt 1 \) a.e. The equation of the form \(f_{\bar{z}}\ =\ \mu(z) f_z \) where \( f_{\bar z}={\bar\partial}f=(f_x+if_y)/2 \), \(f_{z}=\partial f=(f_x-if_y)/2\), \(z=x+iy\), \( f_x \) and \( f_y \) are partial derivatives of the function \(f\) in \(x\) and \(y\), respectively, is said to be a Beltrami equation. The function \(\mu\) is called its complex coefficient, and \( K_{\mu}(z)=\frac{1+|\mu(z)|}{1-|\mu(z)|}\) is called its dilatation quotient. The Beltrami equation is said to be degenerate if \({\rm ess}\,{\rm sup}\,K_{\mu}(z)=\infty\). The existence of homeomorphic solutions in the Sobolev class \(W^{1,1}_{\rm loc}\) has been recently established for many degenerate Beltrami equations under the corresponding conditions on the dilatation quotient \(K_{\mu}\), see e.g. the monograph Gutlyanskii V., Ryazanov V., Srebro U., Yakubov E. The Beltrami equation. A geometric approach. Developments in Mathematics, 26. Springer, New York, 2012 and the further references therein. The main theorem of the paper, Theorem 1, states that a homeomorphic solution \( f:\mathbb D\to\mathbb D \) in the Sobolev class \( W^{1,1}_{\rm loc} \) of the Beltrami equation in the unit disk \(\mathbb D\) has a homeomorphic extension to the boundary that is Hölder continuous if \(K_{\mu}\in L^1(\Bbb D)\) and, for some \(\varepsilon_0\in(0,1)\) and \(C\in[1,\infty)\), $$ \sup\limits_{\varepsilon\in(0,\varepsilon_0)} \int_{\mathbb D\cap D(\zeta,\varepsilon)}K_{\mu}(z) dm(z) \lt C \qquad \forall \zeta \in \partial \mathbb{D} $$ where \(D(\zeta,\varepsilon)=\left\{z\in{\Bbb C}: |z-\zeta| \lt \varepsilon\right\}\).
Abstract It was discovered that there is a formal analogy between Nevanlinna theory and Diophantine approximation. Via Vojta's dictionary, the Second Main Theorem in Nevanlinna theory corresponds to Schmidt's Subspace Theorem in Diophantine approximation. Recently, Cherry, Dethloff, and Tan ( arXiv:1503.08801v2 [math.CV] ) obtained a Second Main Theorem for moving hypersurfaces intersecting projective varieties. In this paper, we shall give the counterpart of their Second Main Theorem in Diophantine approximation.
The paper is devoted to an algebraic analogue of a geometric approach to the classical notion of complex dilatation suggested in the paper arXiv:1701.06259 [math.CV] by the author. At the same time it provides an invariant version of this geometric approach. From the algebraic point of view it is only natural to work with a general field extension K/k of degree 2 instead of the fields of real and complex number (under the assumption that the characteristic is not equal to 2). Given a k-linear map between two K-vector spaces of dimension 1 over K, there are two natural measures of deviation of this map from being K-linear: its conformal dilatation, defined in terms of quadratic forms over k, and its Beltrami form, directly generalizing the classical complex dilatation. It turns out that these two measures are related in the same way as in the classical case. Working with a general field extension does not lead to any new difficulties compared to the classical case, but only clarifies the algebraic aspects of the theory.
Manuel D. Contreras, S. Díaz-Madrigal, P. Gumenyuk
Loewner Theory is a deep technique in Complex Analysis affording a basis for many further important developments such as the proof of famous Bieberbach’s conjecture and well-celebrated Schramm’s stochastic Loewner evolution. It provides analytic description of expanding domains dynamics in the plane. Two cases have been developed in the classical theory, namely the radial and the chordal Loewner evolutions, referring to the associated families of holomorphic self-mappings being normalized at an internal or boundary point of the reference domain, respectively. Recently there has been introduced a new approach (Bracci F et al. in Evolution families and the Loewner equation I: the unit disk. Preprint 2008. Available on ArXiv 0807.1594; Bracci F et al. in Math Ann 344:947–962, 2009; Contreras MD et al. in Loewner chains in the unit disk. To appear in Revista Matemática Iberoamericana; preprint available at arXiv:0902.3116v1 [math.CV]) bringing together, and containing as quite special cases, radial and chordal variants of Loewner Theory. In the framework of this approach we address the question what kind of systems of simply connected domains can be described by means of Loewner chains of chordal type. As an answer to this question we establish a necessary and sufficient condition for a set of simply connected domains to be the range of a generalized Loewner chain of chordal type. We also provide an easy-to-check geometric sufficient condition for that. In addition, we obtain analogous results for the less general case of chordal Loewner evolution considered in (Aleksandrov IA et al. in Complex Analysis. PWN, Warsaw, pp 7–32, 1979; Bauer RO in J Math Anal Appl 302: 484–501, 2005; Goryainov VV and Ba I in Ukrainian Math J 44:1209–1217, 1992).
We study the existence of topologically closed complex curves normalized by bordered Riemann surfaces in complex spaces. Our main result is that such curves abound in any noncompact complex space admitting an exhaustion function whose Levi form has at least two positive eigenvalues at every point outside a compact set, and this condition is essential. The proof involves a lifting method for the boundary of the curve and a newly developed technique of gluing holomorphic sprays over Cartan pairs in Stein manifolds whose value lie in a complex space, with control up to the boundary of the domains. (The latter technique is also exploited in the subsequent papers math.CV/0607185 and math.CV/0609706.) We also prove that any compact complex curve with C^2 boundary in a complex space admits a basis of open Stein neighborhoods. In particular, an embedded disc of class C^2 with holomorphic interior in a complex manifold admits a basis of open polydisc neighborhoods.
A complex manifold Y satisfies the Convex Approximation Property (CAP) if every holomorphic map from a neighborhood of a compact convex set K in a complex Euclidean space C^n to Y can be approximated, uniformly on K, by entire maps from C^n to Y. If X is a reduced Stein space and Z is a stratified holomorphic fiber bundle over X all of whose fibers satisfy CAP, then sections of Z over X enjoy the Oka property with (jet) interpolation and approximation. Previously this has been proved by the author in the case when X is a Stein manifold without singularities (Ann. Math., 163 (2006), 689-707, math.CV/0402278; Ann. Inst. Fourier, 55 (2005), 733-751, math.CV/0411048). We also give existence results for holomorphic sections under certain connectivity hypothesis on the fibers. In the final part of the paper we obtain the Oka property for sections of submersions with stratified sprays over Stein spaces.
This thesis consists of Chapters 1 and 2. The main results are contained in the two preprints and two published papers, listed below. Chapter 1 deals with conformal invariants in the euclidean space Rn; n >= 2; and their interrelation. In particular, conformally invariant metrics and balls of the respective metric spaces are studied. Another theme in Chapter 1 is the study of quasiconformal maps with identity boundary values in two diferent cases, the unit ball and the whole space minus two points. These results are based on the two preprints: R. Klen, V. Manojlovic and M. Vuorinen: Distortion of two point normalized quasiconformal mappings, arXiv:0808.1219[math.CV], 13 pp., V. Manojlovic and M. Vuorinen: On quasiconformal maps with identity boundary values, arXiv:0807.4418[math.CV], 16 pp. Chapter 2 deals with harmonic quasiregular maps. Topics studied are: Preservation of modulus of continuity, in particular Lipschitz continuity, from the boundary to the interior of domain in case of harmonic quasiregular maps and quasiisometry property of harmonic quasiconformal maps. Chapter 2 is based mainly on the two published papers: M. Arsenovic, V. Kojic and M. Mateljevic: On Lipschitz continuity of harmonic quasiregular maps on the unit ball in Rn., Ann. Acad. Sci. Fenn. Math. 33 (2008), no. 1, 315-318. V. Kojic and M. Pavlovic: Subharmonicity of jfjp for quasiregular harmonic functions, with applications, J. Math. Anal. Appl. 342 (2008) 742-746
We prove a theorem of M. Gromov (Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2, 851-897, 1989) to the effect that sections of certain holomorphic submersions h from a complex manifold Z onto a Stein manifold X satisfy the Oka principle, meaning that the inclusion of the space of holomorphic sections into the space of continuous sections is a weak homotopy equivalence. The Oka principle holds if the submersion admits a fiber-dominating spray over a small neighborhood of any point in X. This extends a classical result of Grauert (Holomorphe Funktionen mit Werten in komplexen Lieschen Gruppen, Math. Ann. 133, 450-472, 1957). Gromov's result has been used in the proof of the embedding theorems for Stein manifolds and Stein spaces into Euclidean spaces of minimal dimension (Y. Eliashberg and M. Gromov, Ann. Math. 136, 123-135, 1992; J. Schurmann, Math. Ann. 307, 381-399, 1997). For further extensions see the preprints math.CV/0101034, math.CV/0107039, and math.CV/0110201.
Abstract Applying the Second Main Theorem of [J. Noguchi, J. Winkelmann, K. Yamanoi, The second main theorem for holomorphic curves into semi-Abelian varieties II, Forum Math., in press, e-print archive, math.CV/0405492 ], we deal with the algebraic degeneracy of entire holomorphic curves f : C → X from the complex plane C into a complex algebraic normal variety X of positive log Kodaira dimension that admits a finite proper morphism to a semi-Abelian variety. We will also discuss applications to the Kobayashi hyperbolicity problem.
We give a proof of the following theorem of M. Gromov (Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc., 2 (1989), 851-897). Let Z be a holomorphic fiber bundle over a Stein manifold. If the fiber of Z admits a dominating holomorphic spray then the inclusion of the space of global holomorphic sections of Z into the space of continuous sections is a weak homotopy equivalence. In the classical case when the fiber is a complex Lie group or a homogeneous space this was proved by H. Grauert (Math. Ann., 135 (1958), 263--273). For further results in this direction see the papers math.CV/0101040, math.CV/0101040, math.CV/0101034, math.CV/0101238, math.CV/0107039, math.CV/0110201.
Let X and Y be complex manifolds. One says that maps from X to Y satisfy the Oka principle if the inclusion of the space of holomorphic maps from X to Y into the space of continuous maps is a weak homotopy equivalence. In 1957 H. Grauert proved the Oka principle for maps from Stein manifolds to complex Lie groups and homogeneous spaces, as well as for sections of fiber bundles with homogeneous fibers over a Stein base. In 1989 M. Gromov extended Grauert's result to sections of submersions over a Stein base which admit dominating sprays over small open sets in the base; for proof see [F. Forstneric and J. Prezelj: Oka's principle for holomorphic fiber bundles with sprays, Math. Ann. 317 (2000), 117-154, and the preprint math.CV/0101040]. In this paper we prove the Oka principle for maps from Stein manifolds to any complex manifold Y that admits finitely many sprays which together dominate at every point of Y (such manifold is called subelliptic). The class of subelliptic manifolds contains all the elliptic ones, as well as complements of closed algebraic subvarieties of codimension at least two in a complex projective space or a complex Grassmanian. We also prove the Oka principle for removing intersections of holomorphic maps with closed complex subvarieties A of the target manifold Y, provided that the source manifold is Stein and the manifolds Y and Y\A are subelliptic.
Let z=(z1,�,zn) and , the Laplace operator. A formal power series P(z) is said to be Hessian Nilpotent (HN) if its Hessian matrix is nilpotent. In recent developments in [M. de Bondt, A. van den Essen, A reduction of the Jacobian conjecture to the symmetric case, Proc. Amer. Math. Soc. 133 (8) (2005) 2201�2205. [MR2138860]; G. Meng, Legendre transform, Hessian conjecture and tree formula, Appl. Math. Lett. 19 (6) (2006) 503�510. [MR2170971]. See also math-ph/0308035; W. Zhao, Hessian nilpotent polynomials and the Jacobian conjecture, Trans. Amer. Math. Soc. 359 (2007) 249�274. [MR2247890]. See also math.CV/0409534], the Jacobian conjecture has been reduced to the following so-called vanishing conjecture (VC) of HN polynomials: for any homogeneous HN polynomial P(z) (of degree d=4), we have ?mPm+1(z)=0 for any m0. In this paper, we first show that the VC holds for any homogeneous HN polynomial P(z) provided that the projective subvarieties and of determined by the principal ideals generated by P(z) and , respectively, intersect only at regular points of . Consequently, the Jacobian conjecture holds for the symmetric polynomial maps F=z-P with P(z) HN if F has no non-zero fixed point with . Secondly, we show that the VC holds for a HN formal power series P(z) if and only if, for any polynomial f(z), ?m(f(z)P(z)m)=0 when m0.
In recent works [hep-th/9909147, hep-th/0005259] was found a wonderful correlation between integrable systems and meromorphic functions. They reduce a problem of effictivisation of Riemann theorem about conformal maps to calculation of a string solution of dispersionless limit of the 2D Toda hierarchy. In [math.CV/0103136] was found a recurrent formulas for coeffciens of Taylor series of the string solution. This gives, in particular, a method for calculation of the univalent conformal map from the until disk to an arbitrary domain, described by its harmonic moments. In the present paper we investigate some properties of these formulas. In particular, we find a sufficient condition for convergence of the Taylor series for the string solution of dispersionless limit of 2D Toda hierarchy.
Abstract This paper is the first of a sequence of papers [W. Zhao, Differential operator specializations of noncommutative symmetric functions (submitted for publication). math.CO/0509134 ; W. Zhao, Noncommutative symmetric functions and the inversion problem (submitted for publication). math.CV/0509135 ; W. Zhao, A N CS system over the Grossman–Larson Hopf algebra of labeled rooted trees (submitted for publication). math.CO/0509136 ; W. Zhao, N CS systems over differential operator algebras and the Grossman–Larson Hopf algebra of labeled rooted trees (submitted for publication). math.CO/0509138 . preprint] on the N CS (noncommutative symmetric) systems over differential operator algebras in commutative or noncommutative variables [W. Zhao, Differential operator specializations of noncommutative symmetric functions (submitted for publication). math.CO/0509134 ]; the N CS systems over the Grossman–Larson Hopf algebras [R. Grossman, R.G. Larson, Hopf-algebraic structure of families of trees, J. Algebra 126 (1) (1989) 184–210. [MR1023294]; L. Foissy, Les algebres de Hopf des arbres enracines decores I, II, Bull. Sci. Math. 126 (3) (2002) 193–239; (4) 249–288. See also math.QA/0105212 . [MR1909461]] of labeled rooted trees [W. Zhao, A N CS system over the Grossman–Larson Hopf algebra of labeled rooted trees (submitted for publication). math.CO/0509136 ]; as well as their connections and applications to the inversion problem [H. Bass, E. Connell, D. Wright, The Jacobian conjecture, reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. 7 (1982) 287–330. [MR 83k:14028]; A. van den Essen, Polynomial automorphisms and the Jacobian conjecture, in: Progress in Mathematics, vol. 190, Birkhauser Verlag, Basel, 2000. [MR1790619]] and specializations of NCSFs [W. Zhao, Noncommutative symmetric functions and the inversion problem (submitted for publication). math.CV/0509135 ; W. Zhao, N CS systems over differential operator algebras and the Grossman–Larson Hopf algebra of labeled rooted trees (submitted for publication). math.CO/0509138 . preprint]. In this paper, inspired by the seminal work [I.M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V.S. Retakh, J.-Y. Thibon, Noncommutative symmetric functions, Adv. Math. 112 (2) (1995) 218–348. See also hep-th/9407124 . [MR1327096]] on NCSFs (noncommutative symmetric functions), we first formulate the notion of N CS systems over associative Q -algebras. We then prove some results for N CS systems in general; the N CS systems over bialgebras or Hopf algebras; and the universal N CS system formed by the generating functions of certain NCSFs in [I.M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V.S. Retakh, J.-Y. Thibon, Noncommutative symmetric functions, Adv. Math. 112 (2) (1995) 218–348. See also hep-th/9407124 . [MR1327096]]. Finally, we review some of the main results that will be proved in the following papers [W. Zhao, Differential operator specializations of noncommutative symmetric functions (submitted for publication). math.CO/0509134 ; W. Zhao, A N CS system over the Grossman–Larson Hopf algebra of labeled rooted trees (submitted for publication). math.CO/0509136 ; W. Zhao, N CS systems over differential operator algebras and the Grossman–Larson Hopf algebra of labeled rooted trees (submitted for publication). math.CO/0509138 . preprint] as some supporting examples for the general discussions given in this paper.