Joerg Winkelmann
We discuss an example related to the method of Brody. The example is an open subset of a compact complex torus which is covered by entire curves, but not by Brody curves.
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We study the Kobayashi pseudodistance for orbifolds, proving an orbifold version of Brody's theorem and classifying which one-dimensional orbifolds are hyperbolic.
Rate distortion dimension of random Brody curves
本文提出遍历理论方法研究Brody曲线，证明其不变测度的率失真维数受势函数积分上界约束，并给出取等号的丰富测度，类比Ruelle不等式。
The classical Brody's theorem asserts the equivalence between two notions of hyperbolicity for compact complex spaces, one named after Kobayashi and one expressed in terms of lack of non constant holomorphic entire functions (compactness is only used to prove the harder implication). We extend this theorem to Deligne-Mumford analytic stacks, by first providing definitions of what we think of Kobayashi and Brody hyperbolicity for such objects and then proving the equivalence of these concepts under an assumption of compactness.
A holomorphic map from the complex line to a complex projective space is called normal (a. k. a. Brody curve) if it is uniformly continuous from the Euclidean metric to the Fubini--Study metric. The paper contains a survey of known results about such maps, as well as some new theorems.
In 1981 J.Noguchi proved that in a logarithmic algebraic manifold, having logarithmic irregularity strictly bigger than its dimension, any entire curve is algebraically degenerate. In the present paper we are interested in the case of manifolds having logarithmic irregularity equal to its dimension. We restrict our attention to Brody curves, for which we resolve the problem completely in dimension 2: Theorem: In a logarithmic surface with logarithmic irregularity 2 and logarithmic Kodaira dimension 2, any Brody curve is algebraically degenerate. We also deal with the case of arbitrary logarith…
We establish a Lehto--Virtanen-type theorem and a rescaling principle for an isolated essential singularity of a holomorphic curve in a complex space, which are useful for establishing a big Picard-type theorem and a big Brody-type one for holomorphic curves.
A rescaling principle for quasiregular curves with applications to hyperbolicity
研究拟正则曲线到校准流形的重标度原理，通过引入适应设定的布罗迪双曲性等方法，证明了相关等价性，量化了局部连续性模，回答了问题并构造新例子给出布罗迪双曲性障碍。
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.
In one complex variable it is well known that if we consider the family of all holomorphic functions on the unit disc that fix the origin and with first derivative equal to 1 at the origin, then there exists a constant $ρ$, independent of the functions, such that in the image of the unit disc of any of the functions of the family there is a disc of universal radius $ρ.$ This is the so celebrated Landau's theorem. Many counterexamples to an analogous result in several complex variables exist. In this paper we introduce a class of holomorphic maps for which one can get a Landau's theorem and a B…
Urata showed that a pointed compact hyperbolic variety admits only finitely many maps from a pointed curve. We extend Urata's theorem to the setting of (not necessarily compact) hyperbolically embeddable varieties. As an application, we show that a hyperbolically embeddable variety over a number field $K$ with only finitely many $\mathcal{O}_{L,T}$-points for any number field $L/K$ and any finite set of finite places $T$ of $L$ has, in fact, only finitely many points in any given $\mathbb{Z}$-finitely generated integral domain of characteristic zero. We use this latter result in combination wi…
SKT Hyperbolic and Gauduchon Hyperbolic Compact Complex Manifolds
本文为紧复流形引入SKT双曲性与Gauduchon双曲性，证明前者蕴含Kobayashi/Brody双曲性，后者蕴含除子双曲性，并给出SKT双曲流形万有覆盖上L^2调和空间的消失定理。
A survey on generalizations of Forelli's theorem and related pluripotential methods
本文综述了Forelli解析性定理的推广及相关多重位势方法的最新研究进展。
Chern Holomorphic Sectional Curvature: Canonical Ampleness, Rigidity, and Flatness Counterexamples
本文证明紧致Hermitian流形上常Chern全纯截面曲率的刚性定理：非正曲率情形度量必为Kähler，负曲率时典范丛丰富且为Kähler-Einstein度量，正曲率平衡三维流形等距于带Fubini-Study度量的复射影空间。
In this work, a Vietoris type theorem for the positivity of sine and cosine sum for a particular sequence of real numbers is provided. In this connection, the positivity of a particular type of sine sum involving ratio of some parameters is given, which is new in the literature. Various new results that follow from the Vietoris type theorem include improved estimates for the location of the zeros of a class of trigonometric polynomials and new positive sums for orthogonal polynomials. An open problem is also provided for the partial sums of the generalized polylogarithm.
The Valiron-Titchmarsh theorem on asymptotic behavior of entire functions with negative zeros is extended to subharmonic functions in $\mathbb{R}^n, n\geq 3$. We also show that the existence of the limit $\lim_{\rti} \frac{\log u(r)}{N(r)}$, where $N(r)$ is the averaged counting function of a subharmonic function $u$ with the associated masses on a ray implies the regular asymptotic behavior separately for $u$ and for $N$.
We utilize the Legendre-Fenchel transform and weak geodesics for plurisubharmonic functions to construct a weight function that can be used in the Berndtsson-Lempert method, to give an Ohsawa-Takegoshi extension type of result. Theorem 4.1 and Theorem 0.1 in \cite{OT2017} (Theorem \ref{Theorem A} and \ref{Theorem B} below) follow as two special cases of this result, thus answering affirmatively a question posed by Ohsawa in remark 4.1 in \cite{OT2017}, on the Berndtsson-Lempert method.
Using the three balls inequality, we adapt the elegant ideas of Carleman and Domar from complex analysis to linear elliptic PDE and generalize the classical Levinson's loglog theorem.
We examine when the composition of two entire functions f and g is even, and extend some of our results to cyclic compositions in general. We prove some theorems for the cases when f or g is a polynomial. Two of the key theorems we use are a well-known result of Borel, and a theorem of Baker and Gross concerning when f(p(z))=f(q(z)).
We consider commutative C* -algebras of Toeplitz operators in the weighted Bergman space on the unit ball in $\mathbb{C}^{\mathbf{n}}$. For the algebras of elliptic type we find a new representation, namely as the algebra of operators which are functions of certain collections of commuting unbounded self-adjoint operators in the Bergman space.
A functional inequality related to Domar's uniform boundedness theorem
研究与多马尔一致有界性定理相关的泛函不等式\(f(r + s)\leq g(r)+\alpha f(s)\)，通过给出问题的解，对多马尔定理给出新证明并得到明确界。
On a density problem related to a theorem of Szegő
针对Szegő定理相关的密度问题，即当$\rm{log}(w)∉L^1(\rm{ℝ})$时指数属于子集$Λ⊆ℕ_0$的单项式是否张成$L^2(\rm{ℝ},μ)$，在Olevskii和Ulanovskii针对绝对连续测度结果的基础上，研究了Markoff测度的情况。
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