Complex Variables Papers of the Week

We use a combination of the classical dead-end space method and the interpolation (by bounded holomorphic functions) method to affirmatively solve Corona and Gleason problems for familiar Banach spaces of holomorphic functions such as (Besov-)Lipschitz spaces of holomorphic functions on bounded symmetric domains (including the unit ball and polydisc) in $\mathbb{C}^n$, holomorphic mean (Besov-)Lipschitz spaces, (Besov-)Hardy-Sobolev spaces and some special classes of holomorphic mixed-norm (Bergman) spaces (including weighted Bergman spaces) as well as some special classes of holomorphic (mean) Besov-Lipschitz spaces on the unit ball in $\mathbb{C}^n$.

Let $D\subset\mathbb{C}^n$, $n\ge2$, be a bounded domain that is biholomorphic to a finite-dimensional Teichmüller space. We prove that $\partial D$ has no locally convex boundary patch containing a complex $(n-1)$-dimensional manifold through each of its points.

$ More precisely, we prove that the function \[ \sigma(z)=\frac{1}{\theta(1)-1/3}\sum_{n\geq1}(\theta(n)-1/3)z^{n} \] is not universally convex on the slit domain $\mathbb{C}\backslash[1,\infty).

In general, we show that $K_0$-finite holomorphic functions near $M$ are regular functions on $K$-orbits, and that $M$ carries non-constant holomorphic germs if some $K$-stable prime divisor of $Z$ misses $\kappa$, and, under a connectedness hypothesis, only constants otherwise.

We show that the Lie algebra of algebraic vector fields for any connected, semi-simple matrix group over $\mathbb{F}$, for $\mathbb{F}=\mathbb{R},\mathbb{C}$, can be generated by three $\mathbb{F}$-complete algebraic vector fields.

$ We show that there exists a Jordan curve of positive area whose every parametrization via $\mathbb{S}$ admits a homeomorphic extension that has $p$-Sobolev regularity on $\mathbb{S}^2$ for all exponents $1\le p<2 .

If the function has no zeros, Krzyż's conjecture predicts the stronger bound $2/e$ for every coefficient of positive order. We establish this bound for the seventh coefficient: $|a_7|\leq2/e$, with equality precisely for the functions $\eta\exp[-(1-\xi z^7)/(1+\xi z^7)]$, where $|\eta|=|\xi|=1$.

We study the truncated Dirichlet functional \[ \Phi_n (f) = \sum_{k = 1}^n k |a_k|^2, \quad f(z) = \sum_{k = 0}^\infty a_k z^k, \] when $f$ belongs to the Bloch space.

We study global smooth Calabi flows on polarized Kähler surfaces $(M,L)$, assuming uniformly bounded Sobolev constants and the algebraic conditions $\chi(\mathcal O_M)=1$ and $c_1(M)\cdot c_1(L)>0$.

We study the numerical constructions of almost complex and complex structures on spheres and products of spheres up to real dimension six.

In this paper, we develop a theory of evolution dynamics in a Banach bilateral $\mathbb{H}$-algebra and establish a well-posedness theory for the Cauchy problem of evolution equations with quaternionic variables.

We prove the Edholm--Mathew nonvanishing conjecture for the palindromic polynomials associated with rational Hartogs triangles.

We study spatial Hessian degeneracy along weak Kähler geodesics.

For a bounded weak solution given by the Perron envelope, we prove higher regularity on a smaller spatial domain when the smooth boundary data are locally small.

We prove that $F_{\mathbb{D}}(z)$ is the only GAF whose zeros are determinantal with a Hermitian kernel, up to multiplication by a zero-free holomorphic function and a one-to-one holomorphic change of variable, with a \ non-degeneracy condition imposed to exclude trivial cases.

We consider the binomial differential equation $$ ff'-a(f'')^2=be^{2c},$$ where $a(z)$, $b(z)$, $c(z)$ are polynomials such that $b(z)\not\equiv 0$ and $c(z)$ is non-constant. In this paper, we determine all entire solutions of this equation.

We prove that if a smooth Fano variety $X$ has the same dimension and Picard number as $F$, then the existence of a continuous map $X\to F$ of odd topological degree implies $X$ is isomorphic to $F$.

Given $h_o: \mathbb{S} \to \mathbb{S}$, an orientation-preserving homeomorphism of the unit circle, we seek to establish very weak conditions so as to derive a modulus of continuity for $h_o$ and its inverse $h_o^{-1}$, then use these to derive various compactness properties for families of circle homeomorphisms and solutions to Beltrami equations.

We show that for every $n\ge2$, $p>2$, and positive integer $m$, there exists a finite-volume pseudoconvex domain $D\subset\mathbb C^n$ with real-analytic strongly pseudoconvex boundary whose $p$-Bergman space $A^p(D)$ has dimension $m$.