Abstract: Abstract Holomorphic functions on the maximal Grauert tube of a hyperbolic compact Riemann surface are studied. It is shown that their weighted Bergman spaces are infinite dimensional for arbitrary weight order greater than \(-1\) in spite of the fact that they do not admit any non-constant bounded holomorphic functions. The key ingredient of the proof is a computation of weighted norms of analytic continuations of eigenfunctions of the Laplacian in terms of the hypergeometric function. This result complements our previous work (Adachi in Trans Am Math Soc 374(10):7499–7524, 2021) where it was shown that the space of geodesic segments on the Riemann surface has exactly the same property. PubDate: 2022-04-16

Abstract: Abstract An essentially unique homeomorphic solution to the Beltrami equation with measurable coefficients was found in the 1930s by Morrey. The most well-known proof from the 1960s uses the theory of Calderón–Zygmund and singular integral operators in \(L^p(\mathbb {C})\) . We will present an alternative method to solve the Beltrami equation using the Hodge star operator and standard elliptic PDE theory. We will also discuss a different method to prove the regularity of the solution. This approach is partially based on work by Dittmar. PubDate: 2022-04-12

Abstract: Abstract We consider the set S of possible target dimensions for rational sphere maps whose Hermitian-invariant group is the unitary group. In each source dimension, we show that S is co-finite by applying a classical theorem of Ron Graham on complete polynomial sequences. We establish several results, some computer assisted, finding the largest exceptional value. We close by posing a purely number-theoretic question about these exceptional values. PubDate: 2022-03-09

Abstract: Abstract We present an analytical proof that certain natural metric universal covers are Hadamard metric spaces. If \(\rho \,\mathrm{{d}}s\) induces a complete distance d on a plane domain \(\Omega \) , and \(\rho =\varphi \circ u\) where u is (locally Lipschitz and) subharmonic in \(\Omega \) , \(\varphi \) is positive and increasing on an interval containing \(u(\Omega )\) with \(\log \varphi \) convex, then \((\Omega ,d)\) has a universal cover \(({\tilde{\Omega }},{\tilde{d}})\) which is a Hadamard metric space (with geodesics that have Lipschitz continuous first derivatives). PubDate: 2022-02-16

Abstract: Abstract Curve shortening in the z-plane in which, at a given point on the curve, the normal velocity of the curve is equal to the curvature, is shown to satisfy \(S_tS_z=S_{zz}\) , where S(z, t) is the Schwarz function of the curve. This equation is shown to have a parametric solution from which the known explicit solutions for curve shortening flow; the circle, grim reaper, paperclip and hairclip, can be recovered. PubDate: 2022-02-15

Abstract: Abstract Let \(M= \Gamma \setminus \mathbb {H}_d\) be a compact quotient of the d-dimensional Heisenberg group \(\mathbb {H}_d\) by a lattice subgroup \(\Gamma \) . We show that the eigenvalue counting function \(N^\alpha \left( \lambda \right) \) for any fixed element of a family of second order differential operators \(\left\{ \mathcal {L}_\alpha \right\} \) on M has asymptotic behavior \(N^\alpha \left( \lambda \right) \sim C_{d,\alpha } {\text {vol}}\left( M\right) \lambda ^{d + 1}\) , where \(C_{d,\alpha }\) is a constant that only depends on the dimension d and the parameter \(\alpha \) . As a consequence, we obtain an analog of Weyl’s law (both on functions and forms) for the Kohn Laplacian on M. Our main tools are Folland’s description of the spectrum of \({\mathcal {L}}_{\alpha }\) and Karamata’s Tauberian theorem. PubDate: 2022-02-14

Abstract: Abstract Three hyperbolic-type metrics including the triangular ratio metric, the \(j^*\) -metric, and the Möbius metric are studied in an annular ring. The Euclidean midpoint rotation is introduced as a method to create upper and lower bounds for these metrics, and their sharp inequalities are found. A new Möbius-invariant lower bound is proved for the conformal capacity of a general ring domain by using a symmetric quantity defined with the Möbius metric. PubDate: 2022-02-07

Abstract: Abstract This expository paper provides a view of Maxwell’s equations from the perspective of complex differential forms and the Hodge star operator in \(\mathbb {C}^2\) with respect to the Euclidean and the Minkowski metrics. The electric field and the magnetic field are complex 3-dimensional in this case. The paper shows that holomorphic functions naturally give rise to nontrivial solutions to the equations. A simple necessary and sufficient condition regarding wavelike solutions to the equations is also obtained. In the end, the paper gives an interpretation of the Lorenz gauge condition in terms of the co-differential operator. PubDate: 2022-02-05

Abstract: Abstract We establish a type of the Picard’s theorem for entire curves in \(P^n({\mathbb {C}})\) whose spherical derivative vanishes on the inverse images of hypersurface targets. Then, as a corollary, we prove that there is an union D of finite number of hypersurfaces in the complex projective space \(P^n({\mathbb {C}})\) such that for every entire curve f in \(P^n({\mathbb {C}})\) , if the spherical derivative \(f^{\#}\) of f is bounded on \( f^{-1}(D)\) , then \(f^{\#}\) is bounded on the entire complex plane, and hence, f is a Brody curve. PubDate: 2022-01-19

Abstract: Abstract In this note we establish integral formulas for polyanalytic functions in several variables. More precisely, given a positive integer q, we provide explicit expressions for the reproducing kernels of the weighted Bergman spaces of q-analytic functions on the unit ball in \(\mathbb {C}^n\) and that of q-analytic Fock space in \(\mathbb {C}^n\) . PubDate: 2021-10-07

Abstract: Abstract The exponential Teichmüller spaces \(E_p\) , \(0\le p \le \infty\) , interpolate between the classical Teichmüller space ( \(p=\infty\) ) and the space of harmonic diffeomorphisms \((p=0)\) . In this article, we prove the existence of non-variational critical points for the associated functional: mappings f of the disk whose distortion is p-exponentially integrable, \(0<p<\infty\) , yet for any diffeomorphism g(z) of \({\mathbb {D}}\) with \(g \partial {\mathbb {D}}=identity\) and \(g\ne identity\) we have \(f\circ g\) is not of p-exponentially integrable distortion. PubDate: 2021-08-06

Abstract: Abstract We show that a K-quasiregular \(\omega\) -curve from a Euclidean domain to a Euclidean space with respect to a covector \(\omega\) is locally \((1/K)(\Vert \omega \Vert / \omega _{\ell _1})\) -Hölder continuous. We also show that quasiregular curves enjoy higher integrability. PubDate: 2021-07-22

Abstract: Abstract We consider compact Leviflat homogeneous Cauchy–Riemann (CR) manifolds. In this setting, the Levi-foliation exists and we show that all its leaves are homogeneous and biholomorphic. We analyze separately the structure of orbits in complex projective spaces and parallelizable homogeneous CR-manifolds in our context and then combine the projective and parallelizable cases. In codimensions one and two, we also give a classification. PubDate: 2021-07-15

Abstract: Abstract In this paper we introduce a new distance by means of the so-called Szegő kernel and examine some basic properties and its relationship with the so-called Skwarczyński distance. We also examine the relationship between this distance, and the so-called Bergman distance and Szegő distance. PubDate: 2021-07-13

Abstract: Abstract The systematic study of CR manifolds originated in two pioneering 1932 papers of Élie Cartan. In the first, Cartan classifies all homogeneous CR 3-manifolds, the most well-known case of which is a one-parameter family of left-invariant CR structures on \(\mathrm {SU}_2= S^3\) , deforming the standard ‘spherical’ structure. In this paper, mostly expository, we illustrate and clarify Cartan’s results and methods by providing detailed classification results in modern language for four 3-dimensional Lie groups. In particular, we find that \({\mathrm {SL}_2({\mathbb {R}})}\) admits two one-parameter families of left-invariant CR structures, called the elliptic and hyperbolic families, characterized by the incidence of the contact distribution with the null cone of the Killing metric. Low dimensional complex representations of \({\mathrm {SL}_2({\mathbb {R}})}\) provide CR embedding or immersions of these structures. The same methods apply to all other 3-dimensional Lie groups and are illustrated by descriptions of the left-invariant CR structures for \(\mathrm {SU}_2\) , the Heisenberg group, and the Euclidean group. PubDate: 2021-06-25

Abstract: Abstract Let \(\Omega\) be a bounded open subset of \({\mathbb {R}}^n\) and \(\Delta\) the Euclidean Laplace operator \(\sum _{i=1}^n \partial ^2/\partial x_i^2\) . Let \(\beta (x)\) denote the number of eigenvalues less or equal to x with respect to the eigenvalue problem \(\Delta f = -x f\) on \(\Omega\) with \(f=0\) on the boundary of \(\Omega\) . A well-known result due to Hermann Weyl gives the asymptotic formula \(\beta (x)= (2 \pi )^{-n} B_n m_n(\Omega ) x^{n/2}+ o(x^{n/2})\) as \(x\rightarrow \infty\) , where \(B_n\) is the volume of the unit ball in \({\mathbb {R}}^n\) and \(m_n(\Omega )\) is the volume of \(\Omega\) . In this work, we consider the analogous problem for radial functions in the discrete setting of the homogeneous isotropic tree T of homogeneity \(q+1\) ( \(q\ge 2\) ). As the volume of T with respect to the hyperbolic metric is infinite, we don’t expect and indeed we show that there is no analogous result for the commonly used hyperbolic Laplacian on T. We consider instead the eigenvalue problem for radial functions on T with respect to the Euclidean Laplacian on T introduced in [6], where the boundary condition \(f=0\) means that f converges radially to 0 at \(\infty\) . We prove that \(\beta (x)\) is within 2 of \(\log _q \sqrt{x}\) . We also consider other boundary conditions and pose some open questions. PubDate: 2021-06-10

Abstract: Abstract In this article, we survey the recent literature surrounding the geometry of complex polynomials. Specific areas surveyed are (i) Generalizations of the Gauss–Lucas Theorem, (ii) Geometry of Polynomials Level Sets, and (iii) Shape Analysis and Conformal Equivalence. PubDate: 2021-05-17

Abstract: Abstract Using a recent Mergelyan type theorem for products of planar compact sets, we establish generic existence of universal Taylor series on products of planar simply connected domains \({\varOmega }_i\) , \(i=1,\ldots ,d\) . The universal approximation is realized by partial sums of the Taylor development of the universal function on products of planar compact sets \(K_i\) , \(i=1,\ldots ,d\) such that \({\mathbb {C}}-K_i\) is connected and for at least one \(i_0\) the set \(K_{i_0}\) is disjoint from \({\varOmega }_{i_0}\) . PubDate: 2021-05-15

Abstract: Abstract For a subharmonic function defined in the unit disc, we consider the relation between angular boundary value, angular cluster set, and “angular normality” at a point of the unit circle. PubDate: 2021-04-30