Boundary Behaviour of Conformal Maps

Pommerenke, Ch.

doi:10.1007/978-3-662-02770-7

Cited by 1,443 publications

(569 citation statements)

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“…Hence, the first statement is an immediate consequence of the principle that solutions of ODE depend continuously on the parameters of the ODE. The second statement is an immediate consequence of the Caratheodory kernel theorem (see [35], Theorem 1.8). D PROOF OF THEOREM 3.9.…”

Section: Recognizing the Driving Processmentioning

confidence: 91%

“…1] in the Hausdorff metric.) Indeed, this follows immediately from CaratModory's kernel theorem (see [35], Theorem 1.8) and the fact that the local uniform convergence of conformal maps implies the convergence of the derivatives (by Cauchy's formula for the derivative).…”

Section: Recognizing the Driving Processmentioning

confidence: 94%

“…Some of the basic properties of conformal maps (Riemann's mapping theorem, compactness, Koebe distortion) are also needed for the proof. This material may be learned from the first two chapters of [35], for example. In terms of the theory of conformal mappings, this suffices for understanding the argument showing that the driving process of the LERW converges to Brownian motion.…”

Section: Some Comments About the Proofmentioning

confidence: 99%

“…Suppose v E Z2 n aD and e is an edge incident with v that intersects D. The set of such pairs w = (v, e) will be denoted Va(D). If 1/1: D --+ U is conformal, then 1/I(w) will be shorthand for the limit of 1/f(z) as z --+ v along e (which always exists, by [35], Proposition 2.14).…”

Section: Loewner's Equation and Slementioning

confidence: 99%

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Conformal Invariance Of Planar Loop-Erased Random Walks and Uniform Spanning Trees

Lawler

Schramm

Werner

2011

Selected Works of Oded Schramm

289

477

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This paper proves that the scaling limit of a loop-erased random walk in a simply connected domain D ~ tC is equal to the radial SLE2 path. In particular, the limit exists and is conformally invariant. It follows that the scaling limit of the uniform spanning tree in a Jordan domain exists and is conformally invariant. Assuming that aD is a C 1 -simp1e closed curve, the same method is applied to show that the scaling limit of the uniform spanning tree Peano curve, where the tree is wired along a proper arc A c aD, is the chordal SLEg path in I5 joining the endpoints of A. A by-product of this result is that SLEg is almost surely generated by a continuous path. The results and proofs are not restricted to a particular choice of lattice.

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Section: Recognizing the Driving Processmentioning

confidence: 91%

Section: Recognizing the Driving Processmentioning

confidence: 94%

Section: Some Comments About the Proofmentioning

confidence: 99%

Section: Loewner's Equation and Slementioning

confidence: 99%

See 2 more Smart Citations

Conformal Invariance Of Planar Loop-Erased Random Walks and Uniform Spanning Trees

Lawler

Schramm

Werner

2011

Selected Works of Oded Schramm

289

477

View full text Add to dashboard Cite

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“…It can be shown that an H ∞ function that is continuous and nonvanishing on the closed unit disk is an outer function (see [17], p. 105). Another way of generating outer functions is by considering Smirnov domains (see [13], p. 155). It also can be shown that if the conformal map f maps ∂D onto a rectifiable starlike curve which bounds a Smirnov domain and if log f (e it ) ∈ L 1 , then f is an outer function.…”

mentioning

confidence: 99%

Zeros of cesàro sum approximations

Barnard

Pearce

Wheeler

2001

Complex Variables, Theory and Application: An International Jou

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Abstract.Let D denote the open unit disk and let f (z) = ∞ n=0 anz n be analytic on D with positive monotone decreasing coefficients an. We answer several questions posed by J. Cima on the location of the zeros of polynomial approximants which he originally posed about outer functions. In particular, we show that the zeros of Cesàro approximants to f are well-behaved in the following sense: (1) if an a n+1 → 1, and a 0 am ≤ am b , then ∂D is the only accumulation set for the zeros of the Cesàro sums of f ; and (2) if f has a representation f (z) = ∞ n=0 g 1 n+c z n where g(x) = ∞ n=0 bnx n ≡ 0, bn ≥ 0, then we give sufficient conditions so that the convex hull of the zeros of the Cesàro sums of f will contain D.

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A probabilistic Poisson representation for positive solutions of ?u = u2 in a planar domain

Gall

1997

Comm. Pure Appl. Math.

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We use the stochastic process called the Brownian snake to investigate solutions of the partial differential equation Δu = u2 in a domain D of class C2 of the plane. We prove that nonnegative solutions are in one‐to‐one correspondence with pairs (K, v) where K is a closed subset of ∂D and v is a Radon measure on ∂D\K. Both Kand v are determined from the boundary behavior of the solution u. On the other hand, u can be expressed in terms of the pair (K, v) by an explicit probabilistic representation formula involving the Brownian snake. © 1997 John Wiley & Sons, Inc.

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Boundary Behaviour of Conformal Maps

Cited by 1,443 publications

References 0 publications

Conformal Invariance Of Planar Loop-Erased Random Walks and Uniform Spanning Trees

Conformal Invariance Of Planar Loop-Erased Random Walks and Uniform Spanning Trees

Zeros of cesàro sum approximations

A probabilistic Poisson representation for positive solutions of ?u = u2 in a planar domain

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