Integrability by Compensation and Morse Index Stability for Conformally Invariant Variational Problems in 2D.

After recalling some integrability by compensation results related to conformally invariant Lagrangians in dimension 2, we will present a recent result concerning the upper-semi-continuity of the Morse index plus the nullity of critical points to such variational problems under weak convergence. Precisely we establish that the sum of the Morse indices and the nullity of an arbitrary sequence of weakly converging critical points to a general conformally invariant Lagrangians of maps from an arbitrary closed surface into an arbitrary closed smooth manifold passes to the limit in the following sense : it is asymptotically bounded from above by the sum of the Morse Indices plus the nullity of the weak limit and the bubbles, while it was well known that the sum of the Morse index of the weak limit with the Morse indices of the bubbles is asymptotically bounded from above by the Morse indices of the weakly converging sequence. This is a joint work with Matilde Gianocca and Tristan Rivière