Related papers: Resonances in Loewner equations

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…

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…

A new approach in Loewner Theory proposed by Bracci, Contreras, D\'iaz-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…

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…

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…

We study Liouville-type theorem for polyharmonic H\'enon-Lane-Emden system $(-\Delta)^mu=|x|^av^p,\; (-\Delta)^mv=|x|^bu^q$ when $m,p,q\geq 1, pq\ne 1$, and $a,b\geq 0$. It is a natural conjecture that the nonexistence of positive solutions…

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 $\C^N$.

In this paper, we prove the existence and uniqueness of the solution $f(z,t)$ of the Loewner PDE with normalization $Df(0,t)=e^{tA}$, where $A\in L(X,X)$ is such that $k_+(A)<2m(A)$, on the unit ball of a separable reflexive complex Banach…

A positive linear recurrence sequence is of the form $H_{n+1} = c_1 H_n + \cdots + c_L H_{n+1-L}$ with each $c_i \ge 0$ and $c_1 c_L > 0$, with appropriately chosen initial conditions. There is a notion of a legal decomposition (roughly,…

We prove that any Loewner PDE whose driving term h(z,t) vanishes at the origin, and satisfies the bunching condition r m(Dh(0,t))\geq k(Dh(0,t)) for some r\in R^+, admits a solution given by univalent mappings (f_t: B^q\to C^q). This is…

We find a solution to the Loewner chain equation in the case when the infinitesimal generator satisfies h(0,t)=0, Dh(0,t)=A for any linear operator with m(A)>0. We also study the related classes of spirallike mappings, mappings with…

In this paper we consider the phase retrieval problem for Herglotz functions, that is, solutions of the Helmholtz equation $\Delta u+\lambda^2u=0$ on domains $\Omega\subset\mathbb{R}^d$, $d\geq2$. In dimension $d=2$, if $u,v$ are two such…

Let R be a hyperbolic Riemann surface with boundary $\partial R$ and suppose that $\gamma:[0,T]\to R\cup\partial R$ is a simple curve growing from the boundary of R. By lifting $R_{t}=R\setminus \gamma(0,t]$ to the universal covering space…

We study Loewner chains in $\mathcal{H}_0(\mathbb{D})$ without assuming univalence of each element. We prove a decomposition: every chain admits a factorization $f_t=F\circ g_t$, where $F$ is analytic on $\mathbb{D}(0,r)$ with $r=\lim_{t…

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…

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…

We find all non-rational meromorphic solutions of the equation $ww"-(w')^2=\alpha(z)w+\beta(z)w'+\gamma(z)$, where $\alpha$, $\beta$ and $\gamma$ are rational functions of $z$. In so doing we answer a question of Hayman by showing that all…

In paper found conditions that guarantee that solution of Loewner-Kufarev equation maps unit disc onto domain with quasiconformal rectifiable boundary, or it has continuation on closed unit disc, or it's inverse function has continuation on…

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…

The Loewner equation is known as a one-dimensional reduction of the Benney chain as well as the dispersionless KP hierarchy. We propose a reverse process showing that time splitting in the Loewner or the Loewner-Kufarev equation leads to…