Titles and Abstracts

Andrea Seppi, CMC foliations of quasi-Fuchsian manifolds

Abstract: By a result of Mazzeo-Pacard, every hyperbolic quasi-Fuchsian manifold admits a compact subset whose complement is foliated by constant mean curvature (CMC) surfaces. However, there exist quasi-Fuchsian manifolds that do not admit a global CMC foliation. A conjecture due to Thurston asserts that every almost-Fuchsian manifold has such a global CMC foliation. In this talk I will discuss a partial result in this direction, obtained in a joint work with Diptaishik Choudhury and Filippo Mazzoli: every quasi-Fuchsian manifold in a neighbourhood of the Fuchsian locus is (uniquely and monotonically) foliated by CMC surfaces. Time permitting, in the final part of the talk I will explain how these foliations induce Hamiltonian flows on the cotangent bundle of Teichmüller space.

Giulio Colombo: Minimal graphs over manifolds of non-negative Ricci curvature

Abstract: We present some recent results concerning solutions of the minimal graph equation on complete Riemannian manifolds with non-negative Ricci tensor. This is a natural curvature condition under which one may expect rigidity properties similar to those holding in the Euclidean setting. In this talk we present Liouville and splitting theorems for positive solutions and for non-constant solutions with bounded gradient. We also give sufficient conditions to ensure that solutions with at most linear growth are globally Lipschitz.
This is based on joint works with Eddygledson Souza Gama, Marco Magliaro, Luciano Mari and Marco Rigoli.

Mario B. Schulz: Free boundary minimal surfaces in the unit ball

Abstract: (joint work with Alessandro Carlotto, Giada Franz and David Wiygul).
Free boundary minimal surfaces arise naturally in partitioning problems for convex bodies, in capillarity problems for fluids and in the study of extremal metrics for Steklov eigenvalues on manifolds with boundary. The theory has been developed in various interesting directions, yet many fundamental questions remain open. Two of the most basic ones can be phrased as follows:

(1) Can a surface of any given topology be realised as an embedded free boundary minimal surface in the 3-dimensional Euclidean unit ball? We answer this question affirmatively for surfaces with connected boundary and arbitrary genus.

(2) When they exist, are such embeddings unique up to ambient isometry? We answer this question in the strongest negative terms by providing pairs of non-isometric free boundary minimal surfaces with the same topology and symmetry group.

Debora Impera: Quantitative bounds for the index of translators via topology

Abstract: In this talk I will discuss some quantitative estimate on the extended index of translators for the mean curvature flow in the Euclidean space and with bounded norm of the second fundamental form. The estimate involves the dimension of the space of weighted square integrable $f$-harmonic $1$-forms. I will discuss how the dimension of this space can be related to the topology at infinity of the translator and, in some special situation, we will see how it is possible to obtain quantitative estimates of the stability index in terms of the topology of the surface.

The talk is based on joint works with M. Rimoldi.

Gian Maria dall'Ara: On the geometry of smooth pseudoconvex domains and an uncertainty principle

Abstract: In this talk I will report on the results of my collaboration with S. Mongodi (Università Milano-Bicocca). We recently introduced a new geometrical invariant of the boundary of a smooth pseudoconvex domain sitting in an n-dimensional complex manifold, that we call the Levi core of the domain and which aims at capturing the obstruction to global regularity properties like smoothness of solutions of the d-bar equation. For reasonable domains, the Levi core has complex-analytic structure and carries a natural de Rham cohomology class that has both geometric and analytic significance. I will also discuss work in progress in which we combine the geometry encoded in the Levi core with a "holomorphic uncertainty principle" to shed new light on the classical problem of subellipticity of the d-bar equation.

Carlo Mantegazza: Evolution by curvature of networks in the plane - The state of the art

Abstract: I will present the state of the art of the analysis of the motion by curvature of networks of curves in the plane, discussing what is known and the open problems about existence, uniqueness, singularity formation and asymptotic behavior of the flow.

Andrea Malchiodi: Some existence and regularity results for the Born-Infeld model

Abstract: We consider a degenerate elliptic equation that describes spacelike hypersurfaces in Minkowski's spacetime having prescribed Lorentzian mean curvature, as well as the electric potential in Born-Infeld theory. The problem is variational, and minimizers can be found under rather general assumptions. We provide conditions to guarantee that these weakly solve the required Euler-Lagrange equation, and provide counterexamples as well. This is joint work with Jaeyoung Byeon, Norihisa Ikoma and Luciano Mari.

Francesca Oronzio: Some inequalities involving the ADM mass via linear potential theory

Abstract: In this talk, we describe some new monotonicity formulas holding along the regular level sets of suitable harmonic functions in complete, one–ended asymptotically flat 3–manifolds (M,g), whose topology is sufficiently simple, with nonnegative scalar curvature and with (smooth) minimal, compact, connected boundary if the boundary is present. Using such formulas, we obtain a simple proof of the celebrated positive mass theorem and a mass–capacity inequality. The results discussed are obtained by a collaboration with Virginia Agostiniani, Carlo Mantegazza and Lorenzo Mazzieri.

Barbara Nelli: About a Do Carmo's question on complete stable constant mean curvature hypersurfaces

Abtract: We will do an overview about results related to the following question by Manfredo Perdigão Do Carmo (1989): is a noncompact, complete, stable, constant mean curvature hypersurface of the Euclidean space necessarily minimal?

Alessandra Pluda: Evolution by curvature of networks in the plane - resolution of singularities and stability

Abstract: I will present some recent developments in the analysis of motion by curvature of networks of curves in the plane. In particular I will explain the resolution of singularities and how to continue the flow in a classical PDE framework. Moreover I will show how starting from a suitable Lojasiewicz-Simon inequality it is possible to prove stability of the flow in the sense that a motion by curvature starting from a network sufficiently close in H^2-norm to a minimal one exists for all times and smoothly converges.

Simone Diverio: The Lang conjecture for Kähler hyperbolic manifolds

Abstract: A compact complex manifold is Kobayashi hyperbolic if and only if there is no non constant holomorphic image of the complex plane in it. When the manifold is moreover projective (or, more generally, compact Kähler), Lang conjectured in 1986 that it is Kobayashi hyperbolic if and only if it is of general type (a positivity property of the line bundle of global holomorphic top forms) together with all of its subvarieties.
Lang’s conjecture is still very much open nowadays, but in the last few years it has been proved for some remarkable classes of manifolds, such us compact free quotients of bounded domains or compact Kähler manifolds with negative holomorphic sectional curvature. We shall explain how to verify this conjecture for another interesting class of compact Kähler manifolds, namely Kähler hyperbolic manifolds (which were introduced by M. Gromov in the early ‘90s). The proof involves several different techniques including pluripotential theory, spectral theory for the Hodge Laplacian for non-complete metrics, Ahlfors currents, numerical loci for general big&nef transcendental classes, etc.

This is a joint work with F. Bei, P. Eyssidieux, and S. Trapani.

Giovanni Catino, Some canonical metrics on Riemannian manifolds

Abstract: In this talk I will review recent results concerning the existence of some canonical Riemannian metrics on closed (compact with no boundary) smooth manifolds. The constructions of these metrics are based on Aubin’s local deformations and a variant of the Yamabe problem which was first studied by Gursky.

Giada Franz: Construction of free boundary minimal surfaces and their Morse index

Abstract: A free boundary minimal surface (FBMS) in a given 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.

I will talk about the existence of FBMS, with particular emphasis on the computation of their Morse index (i.e., the number of directions in which the area decreases at second order). Moreover, we will review some existing results relating the Morse index and the topology of a FBMS and investigate some further developments.

Carlo Scarpa: Curvature, stability and deformations of toric manifolds

Abstract: We will give an overview of recent developments in the theory of constant scalar curvature Kähler metrics on toric manifolds, highlighting the relationship between the existence of these metrics and a notion of stability of the moment polytope associated to the manifold. We will then explain how this theory can be extended to study a generalization of the problem that takes into account the additional data of a first-order deformation of the toric manifold, following arXiv:2202.00429. Based on joint work with Jacopo Stoppa.

Matthew J. Gursky, The Dirichlet problem for a fully nonlinear version of the Yamabe problem

Abstract: I will discuss ongoing work with Q. Han, in which we study the problem of conformally deforming the

metric on a manifold with boundary in order to prescribe symmetric functions of the eigenvalues of the Schouten

and Ricci tensors. I will also explain the geometric/topological motivation for this work.