The diffeomorphism group of a closed surface of negative Euler characteristic has contractible components. This is theorem by Earle and Eells (Journal of Differential Geometry 3, 1969). The crucial ingredient for their proof is the solvability of the Beltrami differential equation. Later, Gramain found a purely topological, rather elementary proof of that result but - at least for me - the proof using PDEs is much easier to understand.