Université Libre de Bruxelles, 26th November 2018
“Classical Plateau problem in non-smooth spaces”
In the talk I will discuss a solution of the most classical formulation of the Plateau problem in general metric spaces and applications of arising minimal discs. The solution provides a simplification even in the classical Euclidean setting. The talk will be based on a joint work with Stefan Wenger.
“Finding good parametrizations for metric surfaces”
By the classical uniformization theorem, every smooth Riemann surface is conformally diffeomorphic to a surface of constant curvature. What happens if the smooth Riemannian metric is replaced by a non-smooth distance? Does the so obtained metric surface still admit parametrizations with good geometric and analytic properties? Such questions have been widely studied in the field of Analysis on metric spaces and are important for example in view of applications to Geometric Group Theory. I will show how one can use recently established existence and regularity results for area and energy minimizing discs in metric spaces to obtain canonical parametrizations of metric surfaces. In particular, we obtain a new and conceptually simple proof of a well-known theorem of Bonk and Kleiner on the existence of quasisymmetric parametrizations of linearly locally connected, Ahlfors 2-regular metric 2-spheres. Joint work with Alexander Lytchak.
“Synthetic Ricci lower bounds: new geometric examples”
Singular manifolds appear naturally in geometry when considering quotients of smooth manifolds, their Gromov-Hausdorff limits or geometric flows. An important question in the study of such singular manifolds is to define a relevant notion of curvature, or curvature bounds. The work of Lott-Sturm-Villani and Ambrosio-Gigli-Savaré showed that it is possible to define a curvature-dimension condition on metric measure spaces, that corresponds to a Ricci lower bound in the case of smooth Riemannian manifolds. If some constructions on manifolds (quotients, cones, spherical suspension…) give examples of metric spaces satisfying the curvature-dimension condition, there is not any easy criterion to establish whether the RCD(K,N) condition holds on a manifold with simple singularities. In this talk, we present a geometric criterion for a compact stratified space to satisfy the RCD(K,N) condition: this gives a new large class of examples, including among others manifolds with conical singularities, isolated or not. The talk is based on a joint work with J. Bertrand, C. Ketterer and T. Richard.