Loewner Equation Research Articles

Multiple SLE and the complex Burgers equation

In this note we ask whether one can take the limit of multiple SLE as the number of slits goes to infinity. In the special case of n slits that connect n points of the boundary to one fixed point, one can take the limit of the Loewner equation that describes the growth of those slits in a simultaneous way. In this case, the limit is a deterministic Loewner equation whose vector field is determined by a complex Burgers equation.

The Loewner equation for multiple slits, multiply connected domains and branch points

Let \(\mathbb {D}\subset \mathbb {C}\) be the unit disk and let \(\gamma_{1},\gamma _{2}:[0,T]\to\overline{\mathbb {D}}\setminus\{0\}\) be parametrizations of two slits \(\Gamma_{1}:=\gamma(0,T], \Gamma_{2}:=\gamma_{2}(0,T]\) such that \(\Gamma_{1}\) and \(\Gamma_{2}\) are disjoint.

On tangential slit solution of the Loewner equation

For a non-tangential slit (t), the behavior of the driving function �(t) near zero in the Loewner equation is well understood; for tangential slit, the situation is less clear. In this paper, we investigate the tangential slit p , p > 0, where is a circular arc tangent at 0; p has order p+1 p near zero. Our main result is to give the exact expression of �(t), and its Holder exponent near 0 in terms of p, which has a natural connection with the known results. We also extend this to a general type of tangential slits, and give an estimation of the growth of �(t) near 0.

Regularity of Loewner Curves

The Loewner equation encrypts a growing simple curve in the plane into a real-valued driving function. We show that if the driving function $\lambda$ is in $C^{\beta}$ with $\beta>2$ (or real analytic) then the Loewner curve is in $C^{\beta + \frac{1}{2}}$ (respectively analytic). This is a converse to a result by Earle and Epstein and extends a result of Wong.

Chordal Komatu-Loewner equation and Brownian motion with darning in multiply connected domains

Let D = H ∖ ⋃ k = 1 N C k D={\mathbb H} \setminus \bigcup _{k=1}^N C_k be a standard slit domain where H \mathbb H is the upper half-plane and C k C_k , 1 ≤ k ≤ N 1\leq k\leq N , are mutually disjoint horizontal line segments in H \mathbb {H} . Given a Jordan arc γ ⊂ D \gamma \subset D starting at ∂ H , \partial {\mathbb H}, let g t g_t be the unique conformal map from D ∖ γ [ 0 , t ] D\setminus \gamma [0,t] onto a standard slit domain D t D_t satisfying the hydrodynamic normalization. We prove that g t g_t satisfies an ODE with the kernel on its right-hand side being the complex Poisson kernel of the Brownian motion with darning (BMD) for D t D_t , generalizing the chordal Loewner equation for the simply connected domain D = H . D={\mathbb H}. Such a generalization has been obtained by Y. Komatu in the case of circularly slit annuli and by R. O. Bauer and R. M. Friedrich in the present chordal case, but only in the sense of the left derivative in t t . We establish the differentiability of g t g_t in t t to make the equation a genuine ODE. To this end, we first derive the continuity of g t ( z ) g_t(z) in t t with a certain uniformity in z z from a probabilistic expression of ℑ g t ( z ) \Im g_t(z) in terms of the BMD for D D , which is then combined with a Lipschitz continuity of the complex Poisson kernel under the perturbation of standard slit domains to get the desired differentiability.

Value ranges of univalent self-mappings of the unit disc

We describe the value set {f(z0)|f:D→D univalent,f(0)=0,f′(0)=e−T}, where D denotes the unit disc and z0∈D∖{0}, T>0, by applying Pontryagin's maximum principle to the radial Loewner equation.

Pontryagin’s Maximum Principle for the Loewner Equation in Higher Dimensions

AbstractIn this paper we develop a variational method for the Loewner equation in higher dimensions. As a result we obtain a version of Pontryagin’s maximum principle from optimal control theory for the Loewner equation in several complex variables. Based on recent work of Arosio, Bracci, and Wold, we then apply our version of the Pontryagin maximum principle to obtain first-order necessary conditions for the extremal mappings for a wide class of extremal problems over the set of normalized biholomorphic mappings on the unit ball in ℂn.

Value range of solutions to the chordal Loewner equation

We describe a value range {f(z0)} over the class of holomorphic univalent mappings from H∖K(T) onto the upper half-plane H with the hydrodynamic normalization at infinity where K(T) is an arbitrary hull of half-plane capacity T>0. Without loss of generality choose z0=i. A similar value range over the class of inverse functions f−1 is also described. This is a specification of the result by Roth and Schleissinger.

Loewner chains and Hölder geometry

The Loewner equation provides a correspondence between contin- uous real-valued functions t and certain increasing families of half-plane hulls Kt. In this paper we study the deterministic relationship between specic an- alytic properties of t and geometric properties of Kt. Our motivation comes, however, from the stochastic Loewner equation (SLE ), where the associated function t is a scaled Brownian motion and the corresponding domains HnKt are Holder domains. We prove that if the increasing family Kt is generated by a simple curve and the nal domain HnKT is a Holder domain, then the corresponding driving function has a modulus of continuity similar to that of Brownian motion. Informally, this is a converse to the fact that SLE curves are simple and their complementary domains are Holder, when 4. In the process, we establish general geometric criteria that guarantee that Kt has a Lip(1=2) driving function.

Convergence of an algorithm simulating Loewner curves

The development of Schramm-Loewner evolution (SLE) as the scaling limits of discrete models from statistical physics makes direct simulation of SLE an important task. The most common method, suggested by Marshall and Rohde (MR05), is to sample Brownian motion at discrete times, interpolate appropriately in between and solve explicitly the Loewner equation with this approximation. This algorithm always produces piecewise smooth non self-intersecting curves whereas SLE has been proven to be simple for 2 (0;4), self-touching for 2 (4;8) and space-filling for 8. In this paper we show that this sequence of curves converges to SLE for all 6 8 by giving a condition on deterministic driving functions to ensure the sup-norm convergence of simulated curves when we use this algorithm.

LÖwner evolution and finite-dimensional reductions of integrable systems

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 some known integrable systems.

Constant coefficients in the radial Komatu–Loewner equation for multiple slits

The radial Komatu–Loewner equation is a differential equation for certain normalized conformal mappings that can be used to describe the growth of slits in multiply connected domains. We show that it is possible to choose constant coefficients in this equation in order to generate given disjoint slits and that those coefficients are uniquely determined under a suitable normalization of the differential equation.

Rogosinski's lemma for univalent functions, hyperbolic Archimedean spirals and the Loewner equation

We describe the region V ( z 0 ) of values of f ( z 0 ) for all normalized bounded univalent functions f in the unit disk D at a fixed point z 0 ∈ D . The proof is based on the radial Loewner differential equation. We also prove an analogous result for the upper half-plane using the chordal Loewner equation.

A Komatu–Loewner Equation for Multiple Slits

We give a generalization of the Komatu–Loewner equation to multiple slits. Therefore, we consider an $$n$$ -connected circular slit disk $$\Omega $$ as our initial domain minus $$m\in \mathbb {N}$$ disjoint, simple and continuous curves that grow from the outer boundary $$\partial \mathbb {D}$$ of $$\Omega $$ into the interior. Consequently, we get a decreasing family $$(\Omega _t)_{t\in [0,T]}$$ of domains with $$\Omega _0=\Omega $$ . We will prove that the corresponding Riemann mapping functions $$g_t$$ from $$\Omega _t$$ onto a circular slit disk, which are normalized by $$g_t(0)=0$$ and $$g_t'(0)>0$$ , satisfy a Loewner equation known as the Komatu–Loewner equation.

Driving functions and traces of the Loewner equation

We consider the chordal Loewner differential equation in the upper half-plane, the behavior of the driving function λ(t) and the generated hull Kt when Kt approaches λ(0) in a fixed direction or in a sector. In the case that the hull Kt is generated by a simple curve γ(t) with γ(0) = 0, we prove some sharp relations of \({{\lambda (t)} \mathord{\left/ {\vphantom {{\lambda (t)} {\sqrt t }}} \right. \kern-\nulldelimiterspace} {\sqrt t }}\) and \({{\gamma (t)} \mathord{\left/ {\vphantom {{\gamma (t)} {\sqrt t }}} \right. \kern-\nulldelimiterspace} {\sqrt t }}\) as t → 0 which improve the previous work.

Q-Deformed Loewner Evolution

The Loewner equation, in its stochastic incarnation introduced by Schramm, is an insightful method for the description of critical random curves and interfaces in two-dimensional statistical mechanics. Two features are crucial, namely conformal invariance and a conformal version of the Markov property. Extensions of the equation have been explored in various directions, in order to expand the reach of such a powerful method. We propose a new generalization based on q-calculus, a concept rooted in quantum geometry and non-extensive thermodynamics; the main motivation is the explicit breaking of the Markov property, while retaining scale invariance in the stochastic version. We focus on the deterministic equation and give some exact solutions; the formalism naturally gives rise to multiple mutually-intersecting curves. A general method of simulation is constructed - which can be easily extended to other q-deformed equations - and is applied to both the deterministic and the stochastic realms. The way the $q\neq 1$ picture converges to the classical one is explored as well.

Интегралы уравнения Левнера со степенной управляющей функцией

Интегралы уравнения Левнера со степенной управляющей функцией

Basins of attraction in Loewner equations

We prove that any Loewner PDE on the unit ball $\B^q$ whose driving term $h(z,t)$ vanishes at the origin and satisfies the bunching condition $\ell m(Dh(0,t))\geq k(Dh(0,t))$ for some $\ell\in \mathbb{R}^+$, admits a solution given by univalent mappings $(f_t\colon \B^q\to\C^q)_{t\geq 0}$. This is done by discretizing time and considering the abstract basin of attraction. If $\ell<2$, then the range $\cup_{t\geq 0} f_t(\B^q)$ of any such solution is biholomorphic to $\C^q$.

Scaling limits of anisotropic Hastings–Levitov clusters

We consider a variation of the standard Hastings-Levitov model HL(0), in which growth is anisotropic. Two natural scaling limits are established and we give precise descriptions of the effects of the anisotropy. We show that the limit shapes can be realised as Loewner hulls and that the evolution of harmonic measure on the cluster boundary can be described by the solution to a deterministic ordinary differential equation related to the Loewner equation. We also characterise the stochastic fluctuations around the deterministic limit flow.

Spacefilling curves and phases of the Loewner equation

Similar to the well-known phases of SLE, the Loewner differential equation with Lip(1/2) driving terms is known to have a phase transition at norm 4, when traces change from simple to non-simple curves. We establish the deterministic analog of the second phase transition of SLE, where traces change to space-filling curves: There is a constant C>4 such that a Loewner driving term whose trace is space filling has Lip(1/2) norm at least C. We also provide a geometric criterion for traces to be driven by Lip(1/2) functions, and show that for instance the Hilbert space filling curve and the Sierpinski gasket fall into this class.