  We prove that if two conformal embeddings between Riemann surfaces with
finite topology are homotopic, then they are isotopic through conformal
embeddings. Furthermore, we show that the space of all conformal embeddings in
a given homotopy class deformation retracts into a point, a circle, a torus, or
the unit tangent bundle of the codomain, depending on the induced homomorphism
on fundamental groups. Quadratic differentials play a central role in the
proof.
