Abstract: This talk describes a recent joint work of the speaker with N. Wickramasekera (Cambridge). The work develops a regularity theory, with an associated compactness theorem, for weakly defined hypersurfaces (codimension 1 integral varifolds) of a smooth Riemannian manifold that are stationary and stable on their regular parts for volume preserving ambient deformations. The main regularity theorem gives two structural conditions on such a hypersurface that imply that, away from a set of codimension 7 or higher, the hypersurface is locally either a single smoothly embedded disk or precisely two smoothly embedded disks intersecting tangentially. Easy examples show that neither structural hypothesis can be relaxed. An important special case is when the varifold corresponds to the boundary of a Caccioppoli set, in which case the structural conditions can be considerably weakened. An "effective version" of the compactness theorem has been (a posteriori) established in collaboration with O. Chodosh and N. Wickramasekera.
Abstract: We prove that the space of metrics of bounded geometry and uniformly positive scalar curvature of a given 3-manifold is path-connected, generalizing Marques's result for closed 3-manifolds. This is a joint work with Gérard Besson, Fernando C. Marques and Sylvain Maillot.
Abstract : It is a joint work with G. Courtois, S. Gallot and A. Sambusetti. We prove a compactness theorem for metric spaces with an upper bound on the entropy and other conditions that will be discussed. Several finiteness results will be drawn. It is a work in progress
Abstract: In this talk we present a proof of the log-concavity property of total masses of positive currents on a given compact Kähler manifold, that was conjectured by Boucksom, Eyssidieux, Guedj and Zeriahi. The proof relies on the resolution of complex Monge-Ampère equations with prescribed singularities. As corollary we give an alternative proof of the Brunn-Minkowsky inequality for convex bodies. This is based on a joint work with Tamas Darvas and Chinh Lu.
Abstract : we will explain recent results obtained in collaboration with Andrea Malchiodi, Luca Martinazzi and Pierre-Damien Thizy on equations of the type $\Delta u = \lambda fu e^{u^2}$ with prescribed energy. We are in particular able to obtain existence results in high energies and to compute the total degree of the solutions. This is based on a fine asymptotic analysis of solutions of these equations, locating possible concentration points.
Abstract : In a recent paper, Chodosh and Ketover proved that in an asymptotically Euclidean $3$-manifold there exist properly embedded minimal planes. More precisely, for any point in this manifold, there exists a minimal plane containing that point. In this talk, I will explain how one can also prescribe the unit normal to the minimal plane at that point. Besides, one can show that giving three points there is always a minimal plane containing these three points. This is a joint work with H. Rosenberg.
Abstract: Geometric quantities on two-dimensional Riemannian spheres like the length of the shortest closed geodesic are typically controlled using a minimax argument on the one-cycle space of the sphere, and thus rely on finding short sweep-outs of spheres. In this talk, we highlight a strong analogy between these sweep-outs and the branch decompositions used in structural and algorithmic graph theory. This analogy is then leveraged to improve the known bounds on the size of an optimal sweep-out, and thus on the length of the shortest closed geodesic, as well as establish a duality result between the size of sweep-outs and the diameter of continuous maps.This is joint work with Alfredo Hubard and Francis Lazarus.
Abstract: After a brief introduction to the synthetic notions of Ricci curvature lower bounds in terms of optimal transportation, due to Lott-Sturm-Villani, I will discuss some applications to smooth Riemannian manifolds. In particular I will discuss a quantitative Levy-Gromov inequality and an almost euclidean isoperimetric inequality motivated by the celebrated Perelman’s Pseudo-Locality Theorem for Ricci flow.
Abstract: Given a compact surface with a non-empty boundary, we ask the following question : is there a smooth Riemannian metric wich maximizes the k-th Steklov eigenvalue on this surface ? We will explain the link between this problem and the existence question of minimal surfaces with free boundary in a ball. After stating the first eigenvalue results by Fraser and Schoen, we will address the harder question for higher eigenvalues and we will give an existence result of maximal metrics for metrics lying in a fixed conformal class.
Abstract: An existence theory for min-max minimal surfaces, with arbitrary genus and codimension, was recently devised by Rivière, using a viscous relaxation of the area functional. For any min-max in the space of immersions (from a closed surface to a Riemannian manifold) it produces a 2D stationary varifold, which comes enriched with an additional parametrized structure. We show how this combined structure can be exploited to get the full regularity for the limiting varifold, which is studied axiomatically without any need of stability or codimension assumptions. We also show that the natural counterpart of Marques-Neves multiplicity one conjecture holds in this setting, deducing an upper bound on the Morse index (thanks to a result by Michelat). This is joint work with Tristan Rivière.
Abstract: In the early 80's, Yau conjectured that in any closed $3$-manifold there should be infinitely many minimal surfaces. After introducing the problem and reviewing earlier contributions to the question, I will present a proof of the conjecture. In a first part, I will explain the basics of Almgren-Pitts min-max theory, and some min-max methods developed by F. C. Marques and A. Neves who proved the conjecture when a certain condition, the "Frankel property, is satisfied. I will also outline our strategy of proof in the case when this condition is not satisfied, which is based on a localization of min-max constructions. In a second part, I will focus on the details of the previously mentioned local construction of minimal surfaces in a manifold N with non-empty stable boundary. In order to achieve this, a key element is the construction from N of a non-compact manifold with cylindrical ends. Finally I will show how this local construction is used to prove the conjecture.
Abstract: It is not known whether a contractible 3-manifold admits a complete metric of positive scalar curvature. For example, the Whitehead manifold is a contractible 3-manifold but not homeomorphic to $R^{3}$. In this talk, I will present my proof that it does not have a complete metric with positive scalar curvature. I will further explain that a contractible genus one 3-manifold, a notion introduced by McMillan, does not admit a complete metric of positive scalar curvature.