The modules of a Riemann surface are numerical characteristics (parameters) that are the same for all conformally equivalent Riemann surfaces , which together characterize the conformal equivalence class of a given Riemann surface.
Motivation
A necessary condition for conformal equivalence of two plane domains is the same connection of these regions. According to the Riemann theorem, all simply connected domains with more than one boundary point are conformally equivalent to each other: each such domain can be conformally mapped onto the same canonical domain, which is usually considered as a unit circle. For connected domains n, n> 2, the exact equivalent of the Riemann theorem does not exist: one cannot indicate any fixed domain onto which all domains of a given connected order can be univalently and conformally mapped. This led to a more flexible definition of the canonical n-connected region, which indicates the general geometric structure of this region, but does not fix its modules.
Examples
- conformal classes of compact Riemann surfaces of genus characterized by Valid Modules
- torus ( ) is characterized by two modules;
- -connected flat region, regarded as a Riemann surface with an edge, for characterized by {\ displaystyle 3n-6} modules.
- Each doubly connected area the plane with non-degenerate boundary continua can be conformally mapped onto some circular ring
- , .
- Attitude of the radii of the boundary circles of this ring is a conformal invariant and is called the module of a doubly connected domain .
Literature
- Riemann surface modules - an article from the Mathematical Encyclopedia . G.V. Kuzmina, E. D. Solomentsev