Radó's theorem

In mathematics , various theorems that go back to Tibor Radó are called Radó's theorem. (In addition, there are two on Richard Rado declining tenets , namely the set of Rado in matroid theory and the Rado'schen set in the Ramsey ).

Radó's theorem (Riemann surfaces)

The set of Rado in the theory of Riemann surfaces states that any contiguous Riemann surface the second countable met.

This theorem is a peculiarity of complex 1-dimensional manifolds. The analogous statement in higher dimensions does not apply, which is why the second axiom of countability must be explicitly required in the definition of higher-dimensional complex manifolds .

Radó's Theorem (Harmonic Maps)

It is open , contiguous and convex with a smooth edge . Then there is homeomorphism for each ${\displaystyle \Omega \subset \mathbb {R} ^{2}}$

${\displaystyle f\colon S^{1}\to \partial \Omega }$

a harmonious figure

${\displaystyle u\colon D^{2}\to \Omega }$

with . Here the unit circle disc and its edge denotes . ${\displaystyle u\mid _{S^{1}}=f}$ ${\displaystyle D^{2}}$ ${\displaystyle S^{1}=\partial D^{2}}$

A variant of this Radó theorem, also known as the Radó-Behnke-Stein-Cartan theorem, says:

if a continuous function is analytic , then it is entirely analytic. ${\displaystyle f\colon D^{2}\to \mathbb {C} }$ ${\displaystyle \left\{z\in D^{2}\colon f(z)\not =0\right\}}$ ${\displaystyle D^{2}}$

Beckenbach-Radó theorem (subharmonic functions)

It is an open crowd. Beckenbach-Radó's theorem says that a continuous function is subharmonic if and only if the inequality for all closed spheres ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ ${\displaystyle u\colon \Omega \to \mathbb {R} }$ ${\displaystyle {\overline {B(x,r)}}\subset \Omega }$

${\displaystyle \int _{B(x,r)}u(z)dz\leq \int _{\partial B(x,r)}u(y)dy}$

applies.

Individual evidence

1. ↑ Tibor Radó: On the concept of the Riemann surface , Acta Szeged 2 (2): 101–121 (1925)
2. ↑ Tibor Radó: On a non-continuable Riemann manifold , Math. Z. 20, 1-6 (1924)
3. ↑ Erhard Heinz: An elementary proof of the Radó-Behnke-Stein-Cartan theorem about analytic functions. Math. Ann. 131: 258-259 (1956)
4. ^ Edwin Beckenbach, Tibor Radó: Subharmonic functions and minimal surfaces. Trans. Amer. Math. Soc. 35 (1933), no. 3, 648-661.