Continuation of Carathéodory

This article deals with the continuability of conforming illustrations, for the continuability of dimensions see the dimension extension set from Carathéodory .

In mathematics , Carathéodory's theorem deals with the continuation of angle-preserving images on the edge of their domain of definition.

Conformal illustrations, Riemann illustration theorem

A conformal mapping is by definition a figure that angle gets. A mapping between two subsets of the complex plane is conformable if and only if it is holomorphic or anti-holomorphic and the derivative does not vanish anywhere. ${\displaystyle f\colon U\to V}$

The Riemann mapping theorem states that for every simply connected , open , real subset there is a conforming homeomorphism ${\displaystyle U\subset \mathbb {C} }$

${\displaystyle f\colon U\to D^{2}}$

on the unit disk . It was formulated by Riemann in 1851, but only proved by Carathéodory in 1912 . The Riemann mapping theorem is used, among other things, for the geometry of surfaces. ${\displaystyle D^{2}=\left\{z\in \mathbb {C} :\|z\|<1\right\}}$

The edge of the three areas (yellow, green or purple) is not a Jordan curve, see Lakes of the Wada .

The Riemann mapping theorem is remarkable, among other things, because simply connected, open subsets of the plane can be very complicated, for example its boundary can be a nowhere differentiable , fractal curve of infinite length or even no continuously parameterizable curve at all.

In general, it is not true that the Riemann map becomes a continuous map

${\displaystyle \partial (U)\to S^{1}}$

of the edge can continue on the unit circle . Carathéodory's theorem says, however, that such a continuation exists when the edge is a Jordan curve , i.e. the image of a continuous , injective mapping . This includes non-differentiable, fractal curves, for example the Koch curve . ${\displaystyle \partial (U)}$ ${\displaystyle S^{1}=\left\{z\in \mathbb {C} :\|z\|=1\right\}}$ ${\displaystyle \partial (U)}$ ${\displaystyle S^{1}\to \mathbb {C} }$

Theorem of Carathéodory

A simply connected, open subset of the plane bounded by a Jordan curve.

Theorem : Let it be a simply connected, open subset of the complex plane whose edge is a Jordan curve . Then any conforming mapping ${\displaystyle U\subset \mathbb {C} }$ ${\displaystyle \Gamma :=\partial (U)}$

${\displaystyle f\colon U\to D^{2}}$

steadily towards a homeomorphism of degree ${\displaystyle cl(U)=U\cup \partial (U)}$

${\displaystyle {\overline {f}}\colon cl(U)\to {\overline {D}}^{2}}$

continue on the completed circular disc . In particular is a homeomorphism ${\displaystyle {\overline {D}}^{2}=\left\{z\in \mathbb {C} :\|z\|\leq 1\right\}}$ ${\displaystyle {\overline {f}}\mid _{\Gamma }}$

${\displaystyle {\overline {f}}\mid _{\Gamma }\colon \Gamma \to S^{1}}$.

Conclusion : Every conformal mapping between two simply connected, open subsets of the plane bounded by Jordan curves can be extended to a homeomorphism . ${\displaystyle f\colon U_{1}\to U_{2}}$ ${\displaystyle \Gamma _{1},\Gamma _{2}}$ ${\displaystyle \Gamma _{1}\to \Gamma _{2}}$

reversal

The following statements are equivalent for a bounded , simply connected area : ${\displaystyle U\subset \mathbb {C} }$

• Every conformal mapping can be extended to a homeomorphism . ${\displaystyle f\colon U\to D^{2}}$ ${\displaystyle {\overline {f}}\colon cl(U)\to {\overline {D}}^{2}}$
• The edge of is a Jordan curve. ${\displaystyle U}$
• Each edge point is simple , i.e. H. for each sequence there is a curve with whose picture contains all . ${\displaystyle b\in \partial (U)}$ ${\displaystyle b_{n}\in U,\lim _{n\to \infty }b_{n}=b}$ ${\displaystyle \gamma \colon \left[0,1\right]\to cl(U)}$ ${\displaystyle \gamma (1)=b,\gamma (\left[0,1\right))\in U}$ ${\displaystyle b_{n},n\in N}$

From the equivalence follows: the edge of a restricted, convex area is a Jordan curve. ${\displaystyle U\subset \mathbb {C} }$

Higher-dimensional generalizations

The continuous continuation of mappings on the edge of an open set is a widely ramified research topic in mathematics, see for example the Korevaar-Schoen theorem or the Cannon-Thurston theory .

literature

• Carathéodory, C .: About the mutual relationship of the edges in the conformal mapping of the interior of a Jordanian curve onto a circle. Math. Ann. 73 (1913), no. 2, 305-320.

Web links

• Pommerening: Extension of conformal maps to the boundary

Individual evidence

1. ^ Novinger, WP: An elementary approach to the problem of extending conformal maps to the boundary. Amer. Math. Monthly, 82: 279-282 (1975).