Actually, it can be shown that every compact Riemann surface can be embedded into complex projective 3-space.

Continuing in this vein, compact Riemann surfaces can map to surfaces of lower genus, but not to higher genus, except as constant maps.

In contrast, on a compact Riemann surface every holomorphic function is constant, while there always exist non-constant meromorphic functions.

The result described gives that a very definite meaning, applying to projective curves and compact Riemann surfaces in particular.

It is in constant use in complex analysis, algebraic geometry and complex manifold theory, as the simplest example of a compact Riemann surface.

Another extension of the theory appears in where the underlying space of the Riemann-Hilbert problem is a compact hyperelliptic Riemann surface.

He proved unique ergodicity of horocycle flows on a compact hyperbolic Riemann surfaces in the early 1970s.

In mathematics, a compact Riemann surface is a complex manifold of dimension one that is a compact space.

Note that the uniformization theorem implies that every compact Riemann surface of genus one can be represented as a torus.

Thus, compact Riemann surfaces are characterized topologically simply by their genus.