Cauchy integrals and Möbius geometry of curves

Barrett, David; Bolt, Michael

doi:10.4310/ajm.2007.v11.n1.a6

Cited by 8 publications

(13 citation statements)

References 4 publications

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“…* are parabolas that don't intersect as subsets of D. Their Kerzman-Stein distance is defined to be the angle of intersection of two new parabolas constructed using only the corresponding support line elements [1]. The parabola p 12 is tangent to the first line element at the first point and passes through the second point; the parabola p 21 is tangent to the second line element at the second point and passes through the first point.…”

Section: Vertical Parabolas and Laguerre Transformationsmentioning

confidence: 99%

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Laguerre Arc Length from Distance Functions

Barrett¹,

Bolt

2010

Asian Journal of Mathematics

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Abstract. For the Laguerre geometry in the dual plane, invariant arc length is shown to arise naturally through the use of a pair of distance functions. These distances are useful for identifying equivalence classes of curves, within which the extremal curves are proved to be strict maximizers of Laguerre arc length among three-times differentiable curves of constant signature in a prescribed isotopy class. For smoother curves, it is shown that Laguerre curvature determines the distortion of the distance functions. These results extend existing work for the Möbius geometry in the complex plane.

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Section: Vertical Parabolas and Laguerre Transformationsmentioning

confidence: 99%

“…(Their appearance in Theorem 2 refers to their values on the osculating parabolas at the endpoints.) In earlier work, the authors used these distances to provide estimates for certain Möbius invariant operators that include the Cauchy transform and Kerzman-Stein operator [1]. In §6 we pursue analogous results for invariant operators in D.…”

mentioning

confidence: 97%

Laguerre Arc Length from Distance Functions

Barrett¹,

Bolt

2010

Asian Journal of Mathematics

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“…(This is for a region with continuously differentiable boundary.) It is classical that C and C 0 are bounded on L 2 (bΩ), so evidently the Kerzman-Stein operator A def = C − C * = C 0 − C * 0 is bounded and can be expressed using (1). Notice that the holomorphic and arc length differentials are related via dw = T (w) ds w .…”

Section: Smooth Regions With Large Kerzman-stein Eigenvaluementioning

confidence: 99%

“…Subsequent work on the problem was concerned with giving a complete description of the spectrum for model domains [3], asymptotics of eigenvalues for ellipses with small eccentricity [5], and norm estimates that are invariant with respect to Möbius transformation [1,4]. For a disc or halfplane, there is complete cancellation of singularities and the Kerzman-Stein operator is trivial [13].…”

Section: Theoremmentioning

confidence: 99%

“…In particular, she showed that for regions with continuously differentiable boundary the Kerzman-Stein operator remains compact, and for regions with Lipschitz boundary the Kerzman-Stein equation still holds. At issue is the extent to which regularity of the boundary leads to a cancellation of singularities in the kernel when A is expressed as an integral operator, (1) Af (z) = 1 2πi P.V. Here T (w) ∈ C is the unit tangent vector at w ∈ bΩ and ds = ds w is arc length measure for bΩ.…”

Section: Introductionmentioning

confidence: 99%

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The Kerzman–Stein operator for piecewise continuously differentiable regions

Bolt¹,

Raich²

2014

Complex Variables and Elliptic Equations

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Abstract. The Kerzman-Stein operator is the skew-hermitian part of the Cauchy operator defined with respect to an unweighted hermitian inner product on a rectifiable curve. If the curve is continuously differentiable, the Kerzman-Stein operator is compact on the Hilbert space of square integrable functions; when there is a corner, the operator is noncompact. Here we give a complete description of the spectrum for a finite symmetric wedge and we show how this reveals the essential spectrum for curves that are piecewise continuously differentiable. We also give an explicit construction for a smooth curve whose Kerzman-Stein operator has large norm, and we demonstrate the variation in norm with respect to a continuously differentiable perturbation.

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Conformal Parametrisation of Loxodromes by Triples of Circles

Kisil

Reid

2019

Trends in Mathematics

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We provide a parametrisation of a loxodrome by three specially arranged cycles. The parametrisation is covariant under fractional linear transformations of the complex plane and naturally encodes conformal properties of loxodromes. Selected geometrical examples illustrate the usage of parametrisation. Our work extends the set of objects in Lie sphere geometry-circle, lines and points-to the natural maximal conformally-invariant family, which also includes loxodromes.

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Cauchy integrals and Möbius geometry of curves

Cited by 8 publications

References 4 publications

Laguerre Arc Length from Distance Functions

Laguerre Arc Length from Distance Functions

The Kerzman–Stein operator for piecewise continuously differentiable regions

Conformal Parametrisation of Loxodromes by Triples of Circles

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