Titles and Abstracts

Letizia Branca, Università degli Studi di Milano
Title:  Rigidity of Einstein manifolds with positive Yamabe invariant

Abstract: We provide optimal pinching results on closed Einstein manifolds with positive Yamabe invariant in any dimension, extending the optimal bound for the scalar curvature due to Gursky and LeBrun in dimension four. We also improve the known bounds of the Yamabe invariant \emph{via} the $L^{\frac{n}{2}}$-norm of the Weyl tensor for low-dimensional Einstein manifolds. Finally, we discuss some advances on an algebraic inequality involving the Weyl tensor for dimensions 5 and 6.

Reto Buzano, Università di Torino

Title: Mean curvature flow and mean-convex embeddings

Abstract: In this talk, we will study the space of mean-convex embeddings of spheres and tori into three-dimensional manifolds. After motivating the problem, we first start with a very brief and intuitive overview of mean curvature flow with surgery and explain a gluing construction to topologically undo the surgeries again. We then use this surgery and gluing approach to prove that the moduli space of mean convex two-spheres embedded in complete, orientable 3-dimensional manifolds with nonnegative Ricci curvature is path-connected. This result is sharp in the sense that neither of the conditions of (strict) mean convexity, completeness, and nonnegativity of the Ricci curvature can be dropped or weakened. We next study the number of path components of mean convex Heegaard tori, again in ambient manifolds with nonnegative Ricci curvature. We prove that there are always either one or two path components and this number does not only depend on the homotopy type of the ambient manifold. We give a precise characterization of the two cases and also discuss what happens if the mean convexity condition is weakened to nonnegative mean curvature. This is joint work with Sylvain Maillot building on earlier joint work with Robert Haslhofer and Or Hershkovits.

Davide Dameno, Università degli Studi di Milano
Title: Some canonical metrics via Aubin's local deformations

Abstract: In this talk, we will present some recent results concerning the existence of some canonical Riemannian metrics on closed manifolds: in particular, applying a local deformation method due to Aubin, we show that there exist infinitely many metrics with nowhere vanishing Bach tensor on a closed Riemannian four-manifold. We also show the lack of topological obstructions for the existence of “weak half harmonic Weyl” metrics and discuss some advances concerning the Bach tensor. This is joint work with Giovanni Catino and Paolo Mastrolia.

Elia Fusi, Università di Torino
Title: The homogeneous Generalized Ricci flow

Abstract: The Generalized Ricci flow is the natural analogue of the Ricci flow in the setting of Generalized Geometry. In this talk, I will firstly discuss   motivations for the study of the Generalized Ricci flow and its connection with the pluriclosed flow.  Afterwards, I will focus on  the homogeneous case, describing  how  the Generalized Ricci curvature can be seen as the moment map for a suitable action. Finally,  I will give an interpretation of the generalized Ricci flow as a "bracket flow", in the sense introduced by Lauret,  and discuss long-time behaviour on solvmanifolds and asymptotics in the nilpotent case. This is a joint work  with Ramiro Lafuente and James Stanfield.

Marco Magliaro, Università degli Studi dell'Insubria
Title: Sharp pinching theorems for complete PMC submanifolds in the sphere

Abstract: In 1968 Simons proved that if a compact, minimal submanifold of the unit sphere $f:M^n\to\mathbb S^{n+p}$ has second fundamental form satisfying $|A|^2\le np/(2p-1)$, then either $|A|\equiv0$ and $M$ is a great sphere, or $|A|^2\equiv np/(2p-1)$. Lawson and Chern, do Carmo \& Kobayashi characterized the latter case and proved that if $|A|^2\equiv np/(2p-1)$, then $M$ is a Clifford torus or a Veronese surface. This pinching theorem was later generalized by Alencar \& do Carmo for compact CMC hypersurfaces of the sphere and by Santos for compact PMC submanifolds of the sphere. In this talk we extend the results by Simons, Lawson, Chern, do Carmo \& Kobayashi and Alencar \& do Carmo to complete submanifolds of the sphere. We also partially generalize the result of Santos in dimension $n\leq 6$. This is joint work with L. Mari, F. Roing and A. Savas-Halilaj.

Mario Santilli, Università degli Studi dell'Aquila
Title: Rigidity and compactness of rectifiable boundaries with constant mean curvature in warped product spaces

Abstract: we discuss the extension of well-known rigidity results for constant mean curvature hypersurfaces of Alexandrov (in hyperbolic space) and Brendle (in warped product spaces) to sets of finite perimeter with constant distributional mean curvature.  Joint work with Francesco Maggi.

Marcos Paulo Tassi, Università degli Studi dell'Aquila
Title: Elliptic Weingarten surfaces in R^3 with convex planar boundary

Abstract: A surface $\Sigma$ immersed in $\mathbb{R}^3$ is an elliptic Weingarten surface if its principal curvatures $k_1$ and $k_2$ satisfy an equation of the type $W(k_1,k_2)=0$, for some function $W:\mathbb{R}^2 \rightarrow \mathbb{R}$ of class $C^1$ such that $\frac{\partial W}{\partial k_1}\frac{\partial W}{\partial k_2} > 0$ on $W^{-1}(\{0\})$. Known examples of elliptic Weingarten surfaces include minimal and constant mean curvature surfaces, and surfaces of positive constant gaussian curvature.  In 1996 A. Ros and H. Rosenberg proved that for a strictly convex curve $\Gamma \subset \{z=0\} \subset \mathbb{R}^3$, there exists a constant $h$ depending only on the curve $\Gamma$ such that any compact surface embedded in $\mathbb{R}^3_+:= \{z \geq 0\}$ with constant mean curvature $H \leq h$ must be topologically a closed disk. In this talk we will present a generalization of Ros-Rosenberg Theorem for elliptic Weingarten surfaces in $\mathbb{R}^3$, discussing its proof, which is based on some geometric analysis techniques as the Maximum Principle and the Alexandrov Reflection Method, and the recent classification of elliptic Weingarten surfaces of revolution obtained by I. Fernandez and P. Mira. This is a joint work with B. Nelli and G. Pipoli.