Program, titles and abstracts
Talks:
Title: Morse theory of the distance function and applications
Abstract: We will discuss how the variational theory of the Riemannian distance function of pairs of points, developed along the lines of the Morse-Lusternik-Schnirelmann theory, can be used to define a meaningful notion of "width" of circles embedded in any complete Riemannian manifold, of any dimension. This allow meaningful generalisations of classical notions of plane Euclidean geometry, like the notion of width of convex curves (= least distance between supporting lines) and the notion of curves of constant width. Also, the theory implies the existence of minimising geodesics joining pairs of points in the circle in very particular configurations. The talk will be based on joint work with Rafael Montezuma (UFC - Fortaleza) and Roney Santos (USP - São Paulo).
Title: Title: Annular solutions to the partitioning problem in a ball
Abstract: For any natural number n≥2, we construct a real analytic, one-parameter family of compact embedded CMC annuli with free boundary in the unit ball R3, with a prismatic symmetry group of order 4n. These examples give a negative answer to the uniqueness problem by Nitsche and Wente of whether any annular solution to the partitioning problem in the ball should be rotational. Joint work with Alberto Cerezo and Pablo Mira.
Title: Topological control for min-max free boundary minimal surfaces
Abstract: A free boundary minimal surface (FBMS) in a three-dimensional Riemannian manifold is a critical point of the area functional with respect to variations that constrain its boundary to the boundary of the ambient manifold. A very natural question is the one of constructing FBMS (in a given ambient manifold) of a given topological type.
In this talk, we will focus on one of the methods that have been employed so far to tackle this problem, that is Simon-Smith variant of Almgren-Pitts min-max theory. We will see how this method allows us to control the topology (i.e. genus and number of boundary components) of the resulting surface, and we will present several applications.
Title: Eigenvalue problems and free-boundary minimal surfaces in spherical caps.
Abstract: In this talk, I will present a family of functionals on the space of Riemannian metrics of a compact surface with boundary, defined via eigenvalues of a Steklov-type problem, whose maximizing metrics are induced by free boundary minimal immersions in some geodesic ball of a round sphere. Also, I will discuss the maximizer in the case of a disk and present a characterization of a certain class of annuli. This is joint work with Ana Menezes (Princeton University).
Title: Rigidity of Totally Geodesic Hypersurfaces in Negative Curvature
Abstract: I will talk about some rigidity phenomena involving minimal and totally geodesic hypersurfaces in negatively curved manifolds. I will move from rigidity statements involving curvature, for which the tools include Ricci flow and Ratner's theorems from homogeneous dynamics, to if time permits curvature-free rigidity statements where facts about the geodesic flow in negative curvature become essential.
Title: Balanced metrics, Zoll deformations and isosystolic inequalities in CPn
Title: Rigidity and flexibility of scalar curvature
Abstract: In this talk, I will go through some old and new results concerning the rigidity and flexibility of scalar curvature.
Title: Spectrum of the Laplacian and the Jacobi operator on generalized rotational minimal hypersurfaces of spheres
Title: Blow-up rates and hierarchy structures in sequences of CMC surfaces with bounded Morse index"
Abstract: available here.
Title: The Jenkins-Serrin Theorem in three-manifolds
Abstract: In the sixties, H. Jenkins and J. Serrin proved a famous theorem about minimal graphs in the Euclidean 3-space with infinite boundary values. After reviewing the classical results, we show how to solve the Jenkins-Serrin problem in a 3-manifold with a Killing vector field. This is a joint work with A. Del Prete and J. M. Manzano.
Title: Translation solitons to the mean curvature flow in H3.
Abstract: We present existence and classification results for translation solitons (or simply translators) to the mean curvature flow in the hyperbolic 3-space H3. Joint work with R. de Lima and J. P. dos Santos.
Title: Rigidity of compact quasi-Einstein manifolds with boundary
Abstract: available here.
Title: Minimal surfaces with finite total curvature in H2 × R
Abstract: In the last twenty years, the theory of minimal surfaces in H2 × R has been actively developed. As in the case of the Euclidean space, the examples better understood are those with finite total curvature. Hauswirth and Rosenberg started the study of minimal surfaces with finite total curvature in H2 × R. In this talk we will overview what is known in this theory and we will present the construction of new examples and some classification results.
Title: On the topology and index of minimal/Bryant framed surfaces
Abstract: In this talk we'll discuss a 1-to-1 correspondence between Euclidean minimal and Bryant surfaces, known in the literature as Lawson's correspondence. Along with this correspondence we will describe framed surfaces, which is a class of Euclidean minimal and hyperbolic CMC-1 surfaces that generalize immersed Euclidean minimal surfaces and Bryant surfaces. For this class we prove a lower bound on the (unrestricted) Morse index by a linear function of the genus, number of ends and number of branch points (counting multiplicity), generalizing previous work in the literature. This is based on joint work with Davi Maximo.