University of Grenoble-Alpes

Minimal Lagrangian maps play an important role in Teichmüller theory, with important existence and uniqueness results for hyperbolic surfaces obtained by Labourie, Schoen, Bonsante-Schlenker, Toulisse and others. In positive curvature, it is thus natural to ask whether one can find minimal Lagrangian diffeomorphisms between two spherical surfaces with cone points. In this talk we will show that the answer is negative, unless the two surfaces are isometric. As an application, we obtain a generalization of Liebmann’s theorem for branched immersions of constant curvature in Euclidean space. This is joint work with Christian El Emam.

Abstract: Consider a gaussian field f on R^2 and a level l. One can define a random coloring of the plane by coloring a point x in black if f(x)>-l and in white otherwise. The topology of this coloring is interesting in many respects. One can study the "small scale" topology by counting connected components with fixed topology, or study the "large scale" topology by considering black crossings of large rectangles. I will present results involving these quantities.