Complex Variables

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New submissions for Tue, 28 May 24

[1]  arXiv:2405.15970 [pdf, other]

Title: Bounding deformation spaces of Kleinian groups with two generators

Authors: A. Elzenaar, J. Gong, G.J. Martin, J. Schillewaert

Comments: 12 Figures

Subjects: Complex Variables (math.CV)

In this article we provide simple and provable bounds on the size and shape of the quasiconformal deformation space of the groups $\IZ_p*\IZ_q$, the free product of cyclic groups of order $p$ and $q$, in $\PSL(2,\IC)$ for $3\leq p,q \leq \infty$. Though simple, these bounds are sharp, meeting the highly fractal boundary of the deformation space in four cusp groups. Such bounds have great utility in computer assisted searches for extremal Kleinian groups so as to identify universal constraints (volume, length spectra, etc) on the geometry and topology of hyperbolic $3$-orbifolds. As an application, we prove a strengthened version of a conjecture by Morier-Genoud, Ovsienko, and Veselov on the faithfulness of the specialised Burau representation.

[2]  arXiv:2405.16278 [pdf, ps, other]

Title: Generalized integration operators on analytic tent spaces

Authors: Rong Yang, Lian Hu, Songxiao Li

Subjects: Complex Variables (math.CV); Functional Analysis (math.FA)

In this paper, the boundedness and compactness of generalized integration operators $T_g^{n,k}$ between different analytic tent spaces in the unit disc are completely characterized.

[3]  arXiv:2405.16308 [pdf, other]

Title: n-th Root Optimal Rational Approximants to Functions with Polar Singular Set

Authors: L. Baratchart, H. Stahl, M. Yattselev

Subjects: Complex Variables (math.CV); Classical Analysis and ODEs (math.CA)

Let $ D $ be a bounded Jordan domain and $ A $ be its complement on the Riemann sphere. We investigate the $ n $-th root asymptotic behavior in $ D $ of best rational approximants, in the uniform norm on $ A $, to functions holomorphic on $ A $ having a multi-valued continuation to quasi every point of $ D $ with finitely many branches. More precisely, we study weak$^*$ convergence of the normalized counting measures of the poles of such approximants as well as their convergence in capacity. We place best rational approximants into a larger class of $ n $-th root optimal meromorphic approximants, whose behavior we investigate using potential-theory on certain compact bordered Riemann surfaces.

[4]  arXiv:2405.16313 [pdf, ps, other]

Title: On the problem of Kakeya

Authors: Rados Bakic

Comments: 4 pages

Subjects: Complex Variables (math.CV)

We discuss a form of a well-known problem of Kakeya for complex polynomials. Let p(z) be a complex polynomial. This problem requires to find disc that contains n zeros of some derivative of p(z), provided that location of several zeros of p(z) is known. We find a disc that contains a zero of (k-1)-th derivative, if we know disc that contains k zeros of p(z). This result can be regarded as an extension of the Grace-Heawood theorem where k=2.

[5]  arXiv:2405.16943 [pdf, other]

Title: A counterexample to the weak Shanks conjecture

Authors: Catherine Bénéteau, Dmitry Khavinson, Daniel Seco

Subjects: Complex Variables (math.CV); Classical Analysis and ODEs (math.CA); Functional Analysis (math.FA)

We give an example of a function $f$ non-vanishing in the closed bidisk and the affine polynomial minimizing the norm of $1-pf$ in the Hardy space of the bidisk among all affine polynomials $p$. We show that this polynomial vanishes inside the bidisk. This provides a counterexample to the weakest form of a conjecture due to Shanks that has been open since 1980, with applications that arose from digital filter design. This counterexample has a simple form and follows naturally from [7], where the phenomenon of zeros seeping into the unit disk was already observed for similar minimization problems in one variable.

[6]  arXiv:2405.17232 [pdf, ps, other]

Title: Kähler families of Green's functions

Authors: Vincent Guedj, Tat Dat Tô

Comments: 17 pages

Subjects: Complex Variables (math.CV); Algebraic Geometry (math.AG); Analysis of PDEs (math.AP); Differential Geometry (math.DG)

In a remarkable series of works, Guo, Phong, Song, and Sturm have obtained key uniform estimates for the Green's functions associated with certain K\"ahler metrics. In this note, we broaden the scope of their techniques by removing one of their assumptions and allowing the complex structure to vary. We apply our results to various families of canonical K\"ahler metrics.

[7]  arXiv:2405.17242 [pdf, ps, other]

Title: Modulus of Continuity of Solutions to Complex Monge-Ampère Equations on Stein Spaces

Authors: Guilherme C. Gonçalves

Comments: 13 pages

Subjects: Complex Variables (math.CV); Analysis of PDEs (math.AP); Differential Geometry (math.DG)

In this paper, we study the modulus of continuity of solutions to Dirichlet problems for complex Monge-Amp\`ere equations with $L^p$ densities on Stein spaces with isolated singularities. In particular, we prove such solutions are H\"older continuous outside singular points if the boundary data is H\"older continuous.

[8]  arXiv:2405.17415 [pdf, ps, other]

Title: Splitting aspects of holomorphic distributions with locally free tangent sheaf

Authors: Raphael Constant da Costa

Comments: 23 pages, 2 figures

Subjects: Complex Variables (math.CV)

In this work, we mainly deal with a two-dimensional singular holomorphic distribution $\mathcal{D}$ defined on $M$, in the two situations $M=\mathbb{P}^n$ or $M=(\mathbb{C}^n,0)$, tangent to a one-dimensional foliation $\mathcal{G}$ on $M$, and whose tangent sheaf $T_{\mathcal{D}}$ is locally free. We provide sufficient conditions on $\mathcal{G}$ so that there is another one-dimensional foliation $\mathcal{H}$ on $M$ tangent to $\mathcal{D}$, such that their respective tangent sheaves satisfy the splitting relation $T_{\mathcal{D}}=T_{\mathcal{G}} \oplus T_{\mathcal{H}}$. As an application, we show that if $\mathcal{F}$ is a codimension one holomorphic foliation on $\mathbb{P}^3$ with locally free tangent sheaf and tangent to a nontrivial holomorphic vector field on $\mathbb{P}^3$, then $T_{\mathcal{F}}$ splits. Some results on division of holomorphic differential forms by tangent vector fields are also obtained.

Cross-lists for Tue, 28 May 24

[9]  arXiv:2405.16319 (cross-list from math.FA) [pdf, ps, other]

Title: The complete Pick property for pairs of kernels and Shimorin's factorization

Authors: Scott McCullough, Georgios Tsikalas

Subjects: Functional Analysis (math.FA); Complex Variables (math.CV)

Let $(\mathcal{H}_k, \mathcal{H}_{\ell})$ be a pair of Hilbert function spaces with kernels $k, \ell$. In a 2005 paper, Shimorin showed that a certain factorization condition on $(k, \ell)$ yields a commutant lifting theorem for multipliers $\mathcal{H}_k\to\mathcal{H}_{\ell}$, thus unifying and extending previous results due to Ball-Trent-Vinnikov and Volberg-Treil. Our main result is a strong converse to Shimorin's theorem for a large class of holomorphic pairs $(k, \ell),$ which leads to a full characterization of the complete Pick property for such pairs. We also present an alternative, short proof of sufficiency for Shimorin's condition. Finally, we establish necessary conditions for abstract pairs $(k, \ell)$ to satisfy the complete Pick property, further generalizing Shimorin's work with proofs that are new even in the single-kernel case $k=\ell.$ Shimorin's approach differs from ours in that we do not work with the Nevanlinna-Pick problem directly; instead, we are able to extract vital information for $(k, \ell)$ through Carath\'eodory-Fej\'er interpolation.

Replacements for Tue, 28 May 24

[10]  arXiv:2004.08717 (replaced) [pdf, ps, other]

Title: Hardy-Littlewood theorems and the Bergman distance

Authors: Marijan Markovic

Journal-ref: Annales mathematiques du Quebec 48(2024), 143-156

Subjects: Complex Variables (math.CV)

[11]  arXiv:2405.09012 (replaced) [pdf, ps, other]

Title: An analogue of Koebe's theorem and the openness of a limit map in one class

Authors: Evgeny Sevost'yanov, Valery Targonskii

Subjects: Complex Variables (math.CV)

[12]  arXiv:2211.10886 (replaced) [pdf, ps, other]

Title: Extensions of polynomial plank covering theorems

Authors: Alexey Glazyrin, Roman Karasev, Alexandr Polyanskii

Comments: Lemma 4.1 is corrected after the official publication, fortunately, not affecting the other results

Journal-ref: Bulletin of the London Mathematical Society 56:3 (2024), 1014--1028

Subjects: Metric Geometry (math.MG); Complex Variables (math.CV); Functional Analysis (math.FA)

[13]  arXiv:2311.13420 (replaced) [pdf, ps, other]

Title: Moduli of K3 families over $\mathbb{P}^1$, cycle spaces of IHS period domains, and deformations of complex-hyperkähler metrics

Authors: Daniel Greb, Martin Schwald

Comments: 56 pages, v2: minor changes throughout the paper, added results on coarse moduli spaces for embedded families (Theorem 8.9) and hyperbolicity properties (Section 7.6)

Subjects: Algebraic Geometry (math.AG); Complex Variables (math.CV); Differential Geometry (math.DG)

[14]  arXiv:2402.08076 (replaced) [pdf, other]

Title: A note on double Floquet-Bloch transforms and the far-field asymptotics of Green's functions tailored to periodic structures

Authors: Andrey V. Shanin, Raphael C. Assier, Andrey I. Korolkov, Oleg I. Makarov

Subjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Complex Variables (math.CV)

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