We extend the seminal work of Sacks and Uhlenbeck into the
fractional framework. We look for maps between manifolds M and N that
minimize the critical Sobolev W(s,n/s) energy, where n is the dimension
of the manifold M in the domain. We prove that in the case when the
target manifold N has trivial n-th homotopy group then in every homotopy
class of maps between an n-manifold M to N there exists a minimizing
fractional harmonic map. In case when the n-th homotopy group of the
target manifold is not trivial we prove that there exists a generating
set for the n-th homotopy group of the target manifold N consisting of
minimizing fractional maps.
We develop new tools which are interesting on their own, such as a
removability result for point-singularities and a balanced energy
estimate for non-scaling invariant energies.
Joint work with Armin Schikorra.