26 jun 2019 -- 16:00
Sala Riunioni, DM, Pisa
Abstract.
A celebrated theorem of F. Hélein guarantees that a weakly harmonic map from a two-dimensional domain is always smooth. The proof is of a local nature and assumes that the Dirichlet energy is sufficiently small; under this condition it is possible to re-write the harmonic map equation using a suitably chosen frame which uncovers non-linearities with more favourable regularity properties (so-called div-curl or Wente structures). We will prove a global estimate for harmonic maps without assuming a small energy bound, utilising a powerful theory introduced by T. Rivière. Along the way we will highlight the relevance of Wente-type estimates in neighbouring areas of geometric analysis, and hint as to why the analogous higher-dimensional global estimate remains a challenging open problem. This is a joint work with Tobias Lamm.