Filippo Bracci, Manuel D. Contreras, S. Diaz-Madrigal
We prove that evolution families on complex complete hyperbolic manifolds are in one to one correspondence with certain semicomplete non-autonomous holomorphic vector fields, providing the solution to a very general Loewner type differential equation on manifolds.
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In this paper we introduce a general version of the Loewner differential equation which allows us to present a new and unified treatment of both the radial equation introduced in 1923 by K. Loewner and the chordal equation introduced in 2000 by O. Schramm. In particular, we prove that evolution families in the unit disc are in one to one correspondence with solutions to this new type of Loewner equations. Also, we give a Berkson-Porta type formula for non-autonomous weak holomorphic vector fields which generate such Loewner differential equations and study in detail geometric and dynamical pro…
We characterize infinitesimal generators on complete hyperbolic complex manifolds without any regularity assumption on the Kobayashi distance. This allows to prove a general Loewner type equation with regularity of any order $d\in [1,+\infty ]$. Finally, based on these results, we focus on some open problems naturally arising.
We prove that any Loewner PDE in a complete hyperbolic starlike domain of $\C^N$ (in particular in bounded convex domains) admits an essentially unique univalent solution with values in $\C^N$.
We describe the region $\mathcal{V}(z_0)$ of values of $f(z_0)$ for all normalized bounded univalent functions $f$ in the unit disk $\mathbb{D}$ at a fixed point $z_0 \in \mathbb{D}$. The proof is based on identifying $\mathcal{V}(z_0)$ as the reachable set of the radial Loewner differential equation. We also prove an analogous result for the upper half-plane using the chordal Loewner equation.
We prove that, on a complete hyperbolic domain D\subset C^q, any Loewner PDE associated with a Herglotz vector field of the form H(z,t)=A(z)+O(|z|^2), where the eigenvalues of A have strictly negative real part, admits a solution given by a family of univalent mappings (f_t: D\to C^q) such that the union of the images f_t(D) is the whole C^q. If no real resonance occurs among the eigenvalues of A, then the family (e^{At}\circ f_t) is uniformly bounded in a neighborhood of the origin. We also give a generalization of Pommerenke's univalence criterion on complete hyperbolic domains.
We show that an evolution family of the unit disc is commuting if and only if the associated Herglotz vector field has separated variables. This is the case if and only if the evolution family comes from a semigroup of holomorphic self-maps of the disc.
We study chordal Loewner families in the upper half-plane and show that they have a parametric representation. We show one, that to every chordal Loewner family there corresponds a unique measurable family of probability measures on the real line, and two, that to every measurable family of probability measures on the real line there corresponds a unique chordal Loewner family. In both cases the correspondence is being given by solving the chordal Loewner equation. We use this to show that any probability measure on the real line with finite variance and mean zero has univalent Cauchy transfor…
Loewner chains and evolution families on parallel slit half-planes
本文在复平面有限多连通区域上定义并研究了Loewner链与演化族，发展了一般理论并证明其服从由测度值过程驱动的弦Komatu-Loewner微分方程，方法涉及带有缝合的布朗运动。
Loewner Theory, based on dynamical viewpoint, proved itself to be a powerful tool in Complex Analysis and its applications. Recently Bracci et al [Bracci et al, to appear in J. Reine Angew. Math. Available on ArXiv 0807.1594; Bracci et al, Math. Ann. 344(2009), 947--962; Contreras et al, Revista Matematica Iberoamericana 26(2010), 975--1012] have proposed a new approach bringing together all the variants of the (deterministic) Loewner Evolution in a simply connected reference domain. This paper is devoted to the construction of a general version of Loewner Theory for the annulus launched in [A…
Loewner Theory, based on dynamical viewpoint, is a powerful tool in Complex Analysis, which plays a crucial role in such important achievements as the proof of famous Bieberbach's conjecture and well-celebrated Schramm's Stochastic Loewner Evolution (SLE). Recently Bracci et al [Bracci et al, to appear in J. Reine Angew. Math. Available on ArXiv 0807.1594; Bracci et al, Math. Ann. 344(2009), 947--962; Contreras et al, Revista Matematica Iberoamericana 26(2010), 975--1012] have proposed a new approach bringing together all the variants of the (deterministic) Loewner Evolution in a simply connec…
We characterize regular fixed points of evolution families in terms of analytical properties of the associated Herglotz vector fields and geometrical properties of the associated Loewner chains. We present several examples showing the rôle of the given conditions. Moreover, we study the relations between evolution families and Herglotz vector fields at regular contact points and prove an embedding result for univalent self-maps of the unit disc with a given boundary regular fixed point into an evolution family with prescribed boundary data.
In this paper we introduce a general version of the notion of Loewner chains which comes from the new and unified treatment, given in [arXiv:0807.1594], of the radial and chordal variant of the Loewner differential equation, which is of special interest in geometric function theory as well as for various developments it has given rise to, including the famous Schramm-Loewner evolution. In this very general setting, we establish a deep correspondence between these chains and the evolution families introduced in [arXiv:0807.1594]. Among other things, we show that, up to a Riemann map, such a cor…
One-parameter semigroups of holomorphic functions appear naturally in various applications of Complex Analysis, and in particular, in the theory of (temporally) homogeneous Markov processes. A suitable analogue of one-parameter semigroups in the inhomogeneous setting is the notion of a (reverse) evolution family. In this paper we study evolution families formed by Bernstein functions, which play the role of Laplace exponents for inhomogeneous continuous-state branching processes. In particular, we characterize all Herglotz vector fields that generate such evolution families and give a complex-…
Lowener Theory on Analytic Universal Covering Maps
本文在不假设单叶性的条件下研究Loewner链，证明其可分解为解析映射与经典单叶Loewner链的复合，并建立万有覆盖映射链的Loewner理论，刻画其与区域族核收敛及连通度的关系，进而推广到Fuchsian群并应用于双曲度量。
Among diversity of frameworks and constructions introduced in Loewner Theory by different authors, one can distinguish two closely related but still different ways of reasoning, which colloquially may be described as "increasing" and "decreasing". In this paper we review in short the main types of (deterministic) Loewner evolution discussed in the literature and describe in detail the local duality between "increasing" and "decreasing" cases within the general unifying approach in Loewner Theory proposed recently in [Bracci et al. to appear in J Reine Angew Math; arXiv:0807.1594v1], [Bracci et…
Loewner Theory is a deep technique in Complex Analysis affording a basis for many further important developments such as the proof of famous Bieberbach's conjecture and well-celebrated Schramm's Stochastic Loewner Evolution (SLE). It provides analytic description of expanding domains dynamics in the plane. Two cases have been developed in the classical theory, namely the {\it radial} and the {\it chordal} Loewner evolutions, referring to the associated families of holomorphic self-mappings being normalized at an internal or boundary point of the reference domain, respectively. Recently there h…
We present a new geometric construction of Loewner chains in one and several complex variables which holds on a complete hyperbolic complex manifold M and prove that there is essentially a one-to-one correspondence between evolution families of order d and Loewner chains of the same order. As a consequence we obtain a solution for any Loewner-Kufarev PDE, given by univalent mappings (f_t) from M to a complex manifold N. The problem of finding solutions given by univalent mappings with range in C^n is reduced to investigating whether the union of the images f_t(M) is biholomorphic to a domain i…
Loewner Theory for Bernstein functions II: applications to inhomogeneous continuous-state branching processes
本文通过伯恩斯坦函数和洛维纳理论研究非均匀连续状态分支过程的拉普拉斯指数和生成函数，建立微分方程并探讨其嵌入条件。
A new approach in Loewner Theory proposed by Bracci, Contreras, Díaz-Madrigal and Gumenyuk provides a unified treatment of the radial and the chordal versions of the Loewner equations. In this framework, a generalized Loewner chain satisfies the differential equation $$ \partial_{t}f_{t}(z) = (z - τ(t))(1-\overline{τ(t)}z)\partial_{z}f_{t}(z)p(z,t), $$ where $τ: [0,\infty) \to \overline{\mathbb{D}}$ is measurable and $p$ is called a Herglotz function. In this paper, we will show that if there exists a $k \in [0,1)$ such that $p$ satisfies $$ |p(z,t) - 1| \leq k |p(z,t) + 1| $$ for all $z \in \…
The Loewner equation provides a correspondence between continuous real-valued functions $λ_t$ and certain increasing families of half-plane hulls $K_t$. In this paper we study the deterministic relationship between specific analytic properties of $λ_t$ and geometric properties of $K_t$. Our motivation comes, however, from the stochastic Loewner equation (SLE$_κ$), where the associated function $λ_t$ is a scaled Brownian motion and the corresponding domains $\mathbb{H} \backslash K_t$ are Hölder domains. We prove that if the increasing family $K_t$ is generated by a simple curve and the final d…
Runge embeddings, approximation of biholomorphisms on Stein manifolds, and the Loewner PDE
该研究开发了适用于斯坦流形并集的全纯龙格嵌入逼近延拓原理，将其应用于多类斯坦流形相关问题，推广了现有嵌入结果并构造了洛普纳偏微分方程的解。
Among various methods of finding accessory parameters in the Schwarz-Christoffel integrals, Kufarev's method, based on the Loewner differential equation, plays an important role. It is used for describing one-parameter families of functions that conformally map a canonical domain onto a polygon with a slit the endpoint of which moves along a polygonal line starting from a boundary point. We present a modification of Kufarev's method for the case of several slits, the lengths of which have depend of each other in a certain way. We justify the method and find a system of ODEs describing the dyna…
Epstein curves and holography of the Schwarzian action
本文通过Epstein构造的超曲面证明了Schwarzian作用、Epstein曲线长度与所围面积之间的关系，揭示了Jackiw-Teitelboim引力与Schwarzian场论的膜对偶性，并展示了Epstein曲线构造对共轭轨道的扩展。
We study the Weil-Petersson geometry for holomorphic families of Riemann Surfaces equipped with the unique conical metric of constant curvature -1.
We consider mappings of domains of Riemannian manifolds that admit branch points and satisfy a certain condition regarding the distortion of the modulus of families of paths. We have established logarithmic estimates of distance distortion under such mappings. A separate study relates to the situation when the mappings are defined in metric spaces, and one of them is the Loewner space. We also studied the question of equicontinuity of the families of the indicated mappings in the closure of the domain. In addition, we have established the possibility of continuous extension of these mappings t…
For smooth families with maximal variation, whose general fibers have semi-ample canonical bundle, the generalized Viehweg hyperbolicity conjecture states that the base spaces of such families are of log general type. This deep conjecture was recently proved by Popa-Schnell using the theory of Hodge modules and a theorem by Campana-Păun. In this paper we prove that those base spaces are pseudo Kobayashi hyperbolic, as predicted by the Lang conjecture: any complex quasi-projective manifold is pseudo Kobayashi hyperbolic if it is of log general type. As a consequence, we prove the Brody hyperbol…
In this paper, we prove that the quasi-projective base of any maximally variational smooth family of Calabi-Yau klt pairs is both of log general type, and pseudo Kobayashi hyperbolic. Moreover, such a base is Brody hyperbolic if the family is effectively parametrized.
We prove that quasi-projective base spaces of smooth families of minimal varieties of general type with maximal variation do not admit Zariski dense entire curves. We deduce the fact that moduli stacks of polarized varieties of this sort are Brody hyperbolic, answering a special case of a question of Viehweg and Zuo. For two-dimensional bases, we show analogous results in the more general case of families of varieties admitting a good minimal model.
Conformally Natural Families of Probability Distributions on Hyperbolic Disc with a View on Geometric Deep Learning
本文提出双曲圆盘上一类在保圆盘共形映射群作用下不变的新颖概率分布族，为编码双曲数据不确定性提供便捷模型，并探讨其在几何深度学习、生物信息学及量子物理中的潜在应用。
In this paper, we prove the Kobayashi hyperbolicity of the coarse moduli spaces of canonically polarized or polarized Calabi-Yau manifolds in the sense of complex $V$-spaces (a generalization of complex $V$-manifolds in the sense of Satake). As an application, we prove the following hyperbolic version of Campana's isotriviality conjecture: for the smooth family of canonically polarized or polarized Calabi-Yau manifolds, when the Kobayashi pseudo-distance of the base vanishes identically, the family must be isotrivial, that is, any two fibers are isomorphic. We also prove that for the smooth pr…
In this paper we prove that every quasi-projective base space $V$ of smooth family of minimal projective manifolds with maximal variation is pseudo Kobayashi hyperbolic, i.e. $V$ is Kobayashi hyperbolic modulo a proper subvariety $Z\subsetneq V$. In particular, $V$ is algebraically degenerate, that is, every nonconstant entire curve $f:\mathbb{C}\to V$ has image $f(\mathbb{C})$ which lies in that proper subvariety $Z\subsetneq V$. As a direct consequence, we prove the Brody hyperbolicity of moduli spaces of minimal projective manifolds, which answers a question by Viehweg-Zuo in 2003.
Modulus of elementary domains in the hyperbolic plane
研究双曲四边形和双曲环域内曲线族的模，采用几何分析方法确定其模值。
Volume functions and boundary data of 3-dimensional hyperbolic manifolds
本文综述了凸余紧双曲流形中度量唯一性由边界数据决定及体积函数（如重整化体积）的研究进展，并探讨了与AdS/CFT和Loewner能量的新联系。
We construct families of convex domains that are biholomorphic to bounded domains, but not bounded convex domains. This is accomplished by finding an obstruction related to the Gromov hyperbolicity of the Kobayashi metric.
Hyperbolicity properties of moduli spaces of marked hyperk{ä}hler manifolds
研究标记超kähler流形模空间的双曲性性质，证明Kobayashi伪距离消失，并推广Greb和Schwald的定理到任意维数和非紧曲线情况。
We construct families of hyperbolic hypersurfaces of degree $2n$ in the projective space $\mathbb{P}^n(\mathbb{C})$ for $3 \leq n \leq 6$.
Let F_n be families of entire functions, holomorphically parametrized by a complex manifold M. We consider those parameters in M that correspond to nonescaping-hyperbolic functions, i.e., those maps f in F_n for which the postsingular set P(f) is a compact subset of the Fatou set F(f) of f. We prove that if F_n converge to a family F in the sense of a certain dynamically sensible metric, then every nonescaping-hyperbolic component in the parameter space of F is a kernel of a sequence of nonescaping-hyperbolic components in the parameter spaces of F_n. Parameters belonging to such a kernel do n…
One Parameter Families of Conformal Mappings of Bounded Doubly Connected Polygonal Domains
本文基于参数化Loewner--Komatu方法，提出一种求圆环到有界双连通多边形区域共形映射的近似方法，通过推导描述附加参数与共形模量动力学的偏微分方程组，实现对多条割缝变化的完全控制，并用数值实验验证了方法的有效性。
We construct families of hyperbolic hypersurfaces $X_d\subset\mathbb{P}^{n+1}(\mathbb{C})$ of degree $d\geq {\textstyle{(\frac{n+3}{2})^2}}$.
Marked points of families of hyperbolic automorphisms of smooth complex projective varieties
本文研究正熵自同构族中标记点轨道的稳定性，分别构造几何标准高度与连续势的正闭 $(1,1)$-流，并在曲线情形证明二者一致，还推出 Kawaguchi–Silverman 猜想的特例。
Tropical limit of hyperbolic amoebas of complex analytic surfaces
本文证明了复解析曲面在SL_2(C)中经SU(2)商并缩放后像的收敛性，即双曲阿米巴的热带极限。
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