Riemann Surface - Maps Between Riemann Surfaces

Maps Between Riemann Surfaces

The geometric classification is reflected in maps between Riemann surfaces, as detailed in Liouville's theorem and the Little Picard theorem: maps from hyperbolic to parabolic to elliptic are easy, but maps from elliptic to parabolic or parabolic to hyperbolic are very constrained (indeed, generally constant!). There are inclusions of the disc in the plane in the sphere: but any meromorphic map from the sphere to the plane is constant, any holomorphic map from the plane into the unit disk is constant (Liouville's theorem), and in fact any holomorphic map from the plane into the plane minus two points is constant (Little Picard theorem)!

Read more about this topic:  Riemann Surface

Famous quotes containing the words maps and/or surfaces:

    And at least you know

    That maps are of time, not place, so far as the army
    Happens to be concerned—the reason being,
    Is one which need not delay us.
    Henry Reed (1914–1986)

    But ice-crunching and loud gum-chewing, together with drumming on tables, and whistling the same tune seventy times in succession, because they indicate an indifference on the part of the perpetrator to the rest of the world in general, are not only registered on the delicate surfaces of the brain but eat little holes in it until it finally collapses or blows up.
    Robert Benchley (1889–1945)