In this talk we will present a result concerning the upper-semi-continuity of the Morse index plus the nullity of critical points to general conformally invariant Lagrangians in dimension 2 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

The optimal path to circumnavigate the earth is to follow a great circle, a closed geodesic. For nonround surfaces, the length of such closed geodesics may be arbitrarily small. We show that infinitesimally short geodesics on spherical surfaces produce a quantifiable amount of curvature, measured in terms of the Willmore energy. The energy gap we find is not only optimal but also exclusive to surfaces with genus zero. This is joint work with M. Müller (Leipzig) and C. Scharrer (Bonn).

Identifying any conformally round metric on the 2-sphere with a unique cross section on the standard lightcone in the 3+1-Minkowski spacetime, we gain a new perspective on 2d-Ricci flow on topological spheres. It turns out that in this setting, Ricci flow is equivalent to a null mean curvature flow first studied by Roesch–Scheuer along null hypersurfaces. Exploiting this equivalence, we can translate well-known results from 2d-Ricci flow first proven by Hamilton into a full classification of the singularity models for null mean curvature flow in the Minkowski lightcone. Conversely, we obtain a new proof of Hamilton’s classical result using only the maximum principle.