Abstract: Abstract If \(P(z)=a_n\prod _{j=1}^{n}(z-z_j)\) is a complex polynomial of degree n having all its zeros in \( z \le K, ~ K\ge 1,\) it is known that $$\max _{ z =1} P^{\prime }(z) \ge \frac{2}{1+K^n}\sum _{j=1}^{n}\frac{K}{K+ z_j }\max _{ z =1} P(z) .$$ In this paper, we generalise the above inequality to the polar derivative of a polynomial. Our result includes many other interesting inequalities as special cases. PubDate: 2022-11-23

Abstract: Abstract We construct CR mappings between spheres that are invariant under actions of finite unitary groups. In particular, we combine a tensoring procedure with D’Angelo’s construction of a canonical group-invariant CR mapping to obtain new invariant mappings. We also explore possible gap phenomena in this setting. PubDate: 2022-09-15

Abstract: Abstract This is a survey paper discussing the developments around the so-called finite jet determination problem for CR maps over the past twenty years. PubDate: 2022-09-09

Abstract: Abstract This paper surveys recent progress on the analysis of the \(\Box _b\) operator on quadric submanifolds of \({\mathbb {C}}^n \times {\mathbb {C}}^m \) . We focus our discussion on the (relative) fundamental solution to \(\Box _b\) on quadric submanifolds of arbitrary codimension. We summarize known results regarding \(\Box _b\) -invariance of mappings, necessary and sufficient conditions for solvability and hypoellipticity for \(\Box _b\) , and we describe the \(L^p\) regularity of the complex Green operator on quadrics with nonvanishing Levi form. We discuss the ramifications of these results for many examples of quadrics of codimension 1 and 2. PubDate: 2022-09-09

Abstract: Abstract Cartan’s uniqueness theorem does not hold in general for CR mappings, but it does hold under certain conditions guaranteeing extendibility of CR functions to a fixed neighborhood. These conditions can be defined naturally for a wide class of sets such as local real-analytic subvarieties or subanalytic sets, not just submanifolds. Suppose that V is a locally connected and locally closed subset of \({\mathbb {C}}^n\) such that the hull constructed by contracting analytic discs close to arbitrarily small neighborhoods of a point always contains the point in the interior. Then, restrictions of holomorphic functions uniquely extend to a fixed neighborhood of the point. Using this extension, we obtain a version of Cartan’s uniqueness theorem for such sets. When V is a real-analytic subvariety, we can generalize the concept of infinitesimal CR automorphism and also prove an analogue of the theorem. As an application of these two results we show that, for circular subvarieties, the only automorphisms, CR or infinitesimal, are linear. PubDate: 2022-09-05

Abstract: Abstract In this note, we use scaling principle to study the boundary behaviour of the span metric and its higher-order curvatures on finitely connected Jordan planar domains. A localization of this metric near boundary points of finitely connected Jordan domains is also obtained. Further, we obtain boundary sharp estimates for the span metric near \( C^2 \) -smooth boundary points on such domains. PubDate: 2022-08-27

Abstract: Abstract We survey recent work, published since 2015, on equivariant Oka theory. The main results described in the survey are as follows. Homotopy principles for equivariant isomorphisms of Stein manifolds on which a reductive complex Lie group G acts. Applications to the linearisation problem. A parametric Oka principle for sections of a bundle E of homogeneous spaces for a group bundle \({{\mathscr {G}}}\) , all over a reduced Stein space X with compatible actions of a reductive complex group on E, \({{\mathscr {G}}}\) , and X. Application to the classification of generalised principal bundles with a group action. Finally, an equivariant version of Gromov’s Oka principle based on a notion of a G-manifold being G-Oka. PubDate: 2022-08-24

Abstract: Abstract The existence of a non-defective stationary disc attached to a non-degenerate model quadric in \({\mathbb {C}}^N\) is a necessary condition to ensure the unique 1-jet determination of the lifts of a key family of stationary discs [6]. In this paper, we give an elementary proof of the equivalence when the model quadric is strongly pseudoconvex, recovering a result of Tumanov [14]. Our proof is based on the explicit expression of stationary discs, and opens up a conjecture for the unique 1-jet determination to hold when the model is not necessarily strongly pseudoconvex. PubDate: 2022-08-04

Abstract: Abstract We give a normal form for pseudo-Einstein contact forms and apply it to construct intrinsic CR normal coordinates parametrized by the structure group of CR geometry. The proof is based on the construction of parabolic normal coordinates by Jerison and Lee. PubDate: 2022-08-02

Abstract: Abstract In the present article, we define squeezing function corresponding to polydisk and study its properties. We investigate relationship between squeezing function and squeezing function corresponding to polydisk. PubDate: 2022-07-29

Abstract: Abstract We study the infinitesimal CR automorphisms of polynomial model hypersurfaces of finite multitype, which violates 2-jet determination. We give an exposition of some recent results, which provide explicit description of such “exotic” symmetries in complex dimension three. The results are illustrated by numerous examples. PubDate: 2022-07-06

Abstract: Abstract In this paper, we characterize \(C^2\) -smooth totally geodesic isometric embeddings \(f:\Omega \rightarrow \Omega '\) between bounded symmetric domains \(\Omega \) and \(\Omega '\) which extend \(C^1\) -smoothly over some open subset in the Shilov boundaries and have nontrivial normal derivatives on it. In particular, if \(\Omega \) is irreducible, there exist totally geodesic bounded symmetric subdomains \(\Omega _1\) and \(\Omega _2\) of \(\Omega '\) such that \(f = (f_1, f_2)\) maps into \(\Omega _1\times \Omega _2\subset \Omega \) where \(f_1\) is holomorphic and \(f_2\) is anti-holomorphic totally geodesic isometric embeddings. If \(\text {rank}(\Omega ')<2\text {rank}(\Omega )\) , then either f or \({\bar{f}}\) is a standard holomorphic embedding. PubDate: 2022-06-17

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