The idea of this thesis is to apply the methodology of
geometric heat flows to the study of spaces of diffeomorphisms.
We start by describing the general form that a geometrically
natural flow must take and the implications this has for the
evolution equations of associated geometric quantities. We
discuss the difficulties involved in finding appropriate flows
for the general case, and quickly restrict ourselves to the case
of surfaces. In particular the main result is a global existence,
regularity and convergence result for a geometrically defined
quasilinear flow of maps u between flat surfaces, producing a
strong deformation retract of the space of diffeomorphisms onto a
finite-dimensional submanifold. Partial extensions of this result
are then presented in several directions. For general Riemannian
surfaces we obtain a full local regularity estimate under the
hypothesis of bounds above and below on the singular values of
the first derivative. We achieve these gradient bounds in the
flat case using a tensor maximum principle, but in general the
terms contributed by curvature are not easy to control. We also
study an initial-boundary-value problem for which we can attain
the necessary gradient bounds using barriers, but the delicate
nature of the higher regularity estimate is not well-adapted for
obtaining uniform estimates up to the boundary. To conclude, we
show how appropriate use of the maximum principle can provide a
proof of well-posedness in the smooth category under the
assumption of estimates for all derivatives
