List of talks and abstracts
Alessandro Carlotto
ETH (Switzerland)
Constrained deformations of positive scalar curvature metrics
Abstract. What manifolds support metrics of positive scalar curvature? If so, what can one say about the associated moduli space? These are two fundamental problems in Riemannian Geometry, for which great progress has been made over the last fifty years, but that are still highly elusive and far from being fully resolved. Partly motivated by the study of initial data sets for the Einstein equations in General Relativity, I will present some results that aim at moving one step further, studying the interplay between two different curvature conditions, given by pointwise inequalities on the scalar curvature of a manifold, and the mean curvature of its boundary. In particular, after a broad contextualization, I will focus on recent joint work with Chao Li (Princeton University), where we give a complete topological characterization of those compact 3-manifolds that support Riemannian metrics of positive scalar curvature and mean-convex boundary and, in any such case, we prove that the associated moduli space of metrics is path-connected. We can also refine our methods so to construct continuous paths of non-negative scalar curvature metrics with minimal boundary, and to obtain analogous conclusions in that context as well. In particular, note that we can derive the path-connectedness of asymptotically flat scalar flat Riemannian 3-manifolds with minimal boundary, which goes in the direction of understanding the space of vacuum black-hole solutions to the Einstein field equations. Our work relies on a combination of earlier fundamental contributions by Gromov-Lawson and Schoen-Yau, on the smoothing procedure designed by Miao to handle singular interfaces, and on the interplay of Perelman's Ricci flow with surgery and conformal deformation techniques introduced by Codá Marques in dealing with the closed case.
Marcos P. Cavalcante
Universidade Federal de Alagoas (Brazil)
New index bounds for CMC hypersurfaces
Abstract. We use harmonic forms to prove the index of stability of CMC surfaces in the Euclidian space or in the sphere, is bounded from below by multiple of the topological genus. We also discuss how this technique can be extended to prove that the index of stability of minimal hypersurfaces in the 5-sphere is bounded from below by a multiple of the second Betti number. This is joint work with Darlan de Oliveira.
Alberto Enciso
Instituto de Ciencias Matemáticas (Spain)
Rigidity vs flexibility in nodal sets of eigenfunctions
Abstract. Classical results in spectral geometry show that, while the high-energy behavior of the eigenvalues of, say, the Laplacian on a compact Riemannian manifold is very rigid and determined by Weyl’s law, the low-lying eigenvalues are very flexible. Indeed, by carefully choosing the metric on the manifold one can exactly prescribe the first N eigenvalues of the Laplacian, where N is finite abut arbitrary. In this talk we will explore an analogous flexibility/rigidity phenomenon in the subtler context of nodal sets of eigenfunctions. Time permitting, results about random eigenfunctions will be discussed too. This talk is based on joint work with David Hartley, Daniel Peralta-Salas and Álvaro Romaniega.
Marco Guaraco
Chicago University (USA)
Minimal surfaces and mean curvature flow in quasi-Fuchsian hyperbolic manifolds
Laurent Hauswirth
Université de Marne-la-Vallée (France)
TBA
Abstract. TBA
Yen-Chang Huang
National Taipei University of Nursing and Health Sciences (Taiwan)
Cauchy surface area formula in Cauchy-Riemannian manifolds
Abstract. The concept of surface areas of solid objects is one of the classical research topics for geometry. A more systematic approach was due to the development of geometric measure theory by Lebesgue and Minkowski. In this talk, we focus on the surface areas of objects in Cauchy-Riemannian manifolds (CR manifolds), which can be regarded as the real submanifolds of complex manifolds. After introducing the geometric invariants, some integral formulas related to surface areas will be presented, including Cauchy surface area formula, and we will also show some analogous theorems of Riemannian geometry in CR geometry.
William H. Meeks
University of Massachussets, Amherst (USA)
Complete embedded totally umbilic surfaces in hyperbolic 3-manifolds of finite volume
Abstract. In joint work with Adams and Ramos, we construct for every H in $[0,1)$ and every connected surface $S$ of finite topology and negative Euler characteristic, a complete hyperbolic 3-manifold $M(S,H)$ of finite volume such that $M(S,H)$ contains a proper two-sided totally umbilic embedding of S with mean curvature $H$. Conversely, we prove that any complete embedded totally umbilic surface with constant mean curvature $H$ in $[0,1)$ in a hyperbolic 3-manifold $X$ of finite volume has finite nonpositive Euler Characteristic and it must be properly embedded in $X$. Moreover, no such $X$ contains a connected properly embedded, noncompact constant mean curvature surface with constant mean curvature greater than or equal to 1.
Pablo Mira
Universidad P. de Cartagena (Spain)
Weingarten spheres in homogeneous three-manifolds
Abstract. The Abresch-Rosenberg theorem shows that any CMC sphere immersed in a rotationally symmetric homogeneous three-manifold $M$ is a sphere of revolution. In this talk we will extend this result to immersed spheres in $M$ that satisfy any smooth elliptic relation $\Phi(H,K_e,K)=0$ between its mean, extrinsic and intrinsic curvatures, where $\Phi$ is subject to the following condition: the unique inextendible rotational surface $S$ in $M$ that satisfies this equation and touches its rotation axis orthogonally has bounded second fundamental form. As consequences, we prove: (i) any elliptic Weingarten sphere immersed in $\mathbb{H}^2\times\mathbb{R}$ is a rotational sphere. (ii) Any sphere of constant positive extrinsic curvature immersed in $M$ is a rotational sphere. (Joint work with J.A. Gálvez).
Antonio Ros
Universidad de Granada (Spain)
Index one minimal surfaces in positively curved 3-manifolds
Abstract. Let $M$ be a compact orientable riemannian 3-manifold with positive Ricci curvature and let $\Sigma\subset M$ be a closed orientable minimal surface of index one. The second variation formula of the area implies the genus of $\Sigma$ is $g\leq 3$ and min-max area theory guarantees the existence of such a surface with $g\leq 2$ for all $M$. At the present the meaning of the option $g=3$ is not well understood. In this talk we review some related results and we construct a closed minimal surface $\Sigma\subset M$ of index one and genus three.
Pieralberto Sicbaldi
Universidad de Granada (Spain)
Existence and regularity of Faber-Krahn minimizers in a Riemannian manifold
Abstract. We consider the problem of finding domains that minimize the first eigenvalue of the Dirichlet Laplacian in a Riemannian manifold under volume constraint (Faber-Krahn minimizers). In the Euclidean setting such domains are balls, and existence and regularity of such domains is trivial. In a non-Euclidean setting very few results are known. In this talk we will show a general result of existence and regularity of Faber-Krahn minimizers, inspired by the analogous result of existence and regularity of the solutions of the isoperimetric problems in a Riemannian manifold. In particular we will show that Faber-Krahn minimizers are regular till dimension 5, and that there exists a critical dimension between 5 and 7 after which they can have singularities. Such critical dimension is related to the Alf-Caffarelli cone. This is a joint work with J. Lamboley.
Jonathan Zhu
Princeton University (USA)
Mean convex mean curvature flow with free boundary
Abstract. Notions of non-collapsing for mean convex mean curvature flow have important consequences for the regularity and structure of such flows. We’ll discuss the generalisation of these results to the free boundary setting, including ongoing joint work with N. Edelen, R. Haslhofer and M. Ivaki.