The exponential map of the group of area-preserving diffeomorphisms of a   surface with boundary
  We prove that the Riemannian exponential map of the right-invariant $L^2$ metric on the group of volume-preserving diffeomorphisms of a two-dimensional manifold with a nonempty boundary is a nonlinear Fredholm map of index zero. 
James Benn, Gerard Misiolek, Stephen C. Preston
math.DG