Abstract
In this chapter, we study regularity properties curves in the capacity parametrization and their convergence with respect to the uniform norm on compact time intervals.
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Notes
- 1.
It is possible to give a more quantitative estimates for this convergence using the harmonic measure.
- 2.
Remember that constant driving term corresponds to the trivial Loewner chain, which is a straight vertical line segment.
- 3.
A boundary point is accessible, if there is a Jordan arc in the domain ending at that point. If we apply a conformal map from the domain onto \(\mathbb {D}\), say, then the image of that arc is continuous up to the boundary. Consequently, the accessible point is always a limit along the image of a Jordan arc in \(\mathbb {D}\) under a conformal map from \(\mathbb {D}\) onto the domain. The radial limit of the conformal map at the same boundary point of \(\mathbb {D}\) exists and is equal to the other limit as follows from Corollary 2.17 of [9].
- 4.
A careful reader can notice that \(E^*\) defined in the latter way, which is more general, doesn’t necessarily define a (simple) graph, but a multigraph where a pair of vertices can be linked by several edges and where the endpoints of an edge don’t need to be distinct vertices.
- 5.
It is natural to select the edge that belongs to \(\partial \varOmega _\delta \) if that exists. If it doesn’t exist, we can always add such an edge to \(\partial \varOmega _\delta \) without disturbing any of the required properties.
- 6.
Such a curve is called a crossing and a crossing that doesn’t contain a proper subcrossing is called a minimal crossing .
- 7.
A small calculation shows that the long side of the rhombi has length \((2m-1)n\) and the short side \((m-1)n\).
- 8.
The minimum number of crossings is finite since there are even smooth crossings such as any “radial” path \(t \mapsto \phi (z_0 + t e^{i \theta })\), \(t \in (r,R)\) and \(\theta \in \mathbb {R}\).
- 9.
That is, define \([\gamma ] = \{ \gamma \circ \phi \,:\, \phi : [0,1] \rightarrow [0,1] \text { increasing homeomorphism}\}\).
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Kemppainen, A. (2017). Regularity and Convergence of Random Curves. In: Schramm–Loewner Evolution. SpringerBriefs in Mathematical Physics, vol 24. Springer, Cham. https://doi.org/10.1007/978-3-319-65329-7_6
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