Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature

In this paper we consider complete noncompact Riemannian manifolds $(M, g)$ with nonnegative Ricci curvature and Euclidean volume growth, of dimension $n \geq 3$. We prove a sharp Willmore-type inequality for closed hypersurfaces $\partial \Omega$ in $M$, with equality holding true if and only if $(M{\setminus}\Omega, g)$ is isometric to a truncated cone over $\partial\Omega$. An optimal version of Huisken's Isoperimetric Inequality for $3$-manifolds is obtained using this result. Finally, exploiting a natural extension of our techniques to the case of parabolic manifolds, we also deduce an enhanced version of Kasue's non existence result for closed minimal hypersurfaces in manifolds with nonnegative Ricci curvature.Comment: Any comment is welcome

Hands on Workshops. ENCODE report on digital competences, learning outcomes and best practices in teaching and learning

The report presents the results of the feedback and competence questionnaires distributed during workshops/training activities organized by ENCODE or associated partners. These results are useful to identify eventual learning needs, measure the improvement in digital competences and design teaching materials and programmes of next training events. The present analysis is based on data collected by the following events (in chronological order): - the “Epigrafia digitale e EpiDoc Epigrafia greca” Workshop, held by A. Bencivenni and I. Vagionakis within the Greek Epigraphy Class of the MA in Classics/Ancient History/Archaeology of the University of Bologna (October, 12th-14th, 2020) - the “ENCODE Greek and Latin Epigraphy Workshop”, organized by the Department of History and Cultures of the University of Bologna, part of the first Multiplier Event of the ENCODE Project (January, 26th-29th, 2021) - the “EpiDoc Workshop London/Bologna”, organized by G. Bodard (Institute of Classical Studies, University of London) and I. Vagionakis (Department of History and Cultures, University of Bologna), held on April, 12th- 16th, 2021 - the “Edizioni digitali di testi sanscriti: introduzione a xml e tei” Workshop, organized by G. Buriola, M. Franceschini, I. Vagionakis (Department of History and Cultures, University of Bologna), held on April, 26th- 29th, 2021 - the “Linked Open Data for Written Artefacts Intensive Training”, organized by the Hiob Ludolf Centre for Ethiopian Studies of the University of Hamburg, part of the second Multiplier Event of the ENCODE project (May, 26th-28th, 2021) - the “Training Workshop Multilingual and Multicultural Digital Infrastructures for Ancient Written Artefacts”, organized by the Department of Ancient History of the Katholieke Universiteit Leuven, part of the third Multiplier Event (November, 3rd-5th, 2021) - the “ENCODE Winter School Papyrology for non-specialists” organized by the Institut für klassische Philologie of the Julius-Maximilians-Universität of Würzburg, part of the fourth Multiplier Event (February, 14th- 17th, 2022).The ENCODE Project (KA2-2020-1-IT02-KA203-079585) was financed by the European Commission in the framework of the Erasmus+ Strategic partnership for higher education

Nonlinear isocapacitary concepts of mass in nonnegative scalar curvature

We deal with suitable nonlinear versions of Jauregui's Isocapacitary mass in $3$-manifolds with nonnegative scalar curvature and compact outermost minimal boundary. These masses, which depend on a parameter $1<p\leq 2$, interpolate between Jauregui's mass $p=2$ and Huisken's Isoperimetric mass, as $p \to 1^+$. We derive Positive Mass Theorems for these masses under mild conditions at infinity, and we show that these masses do coincide with the $\mathrm{ADM}$ mass when the latter is defined. We finally work out a nonlinear potential theoretic proof of the Penrose Inequality in the optimal asymptotic regime

The equality case in the substatic Heintze-Karcher inequality

We provide a rigidity statement for the equality case for the Heintze-Karcher inequality in substatic manifolds. We apply such result in the warped product setting to fully remove assumption (H4) in the celebrated Brendle's characterization of constant mean curvature hypersurfaces in warped products.Comment: 19 pages. Comments welcom

On the Isoperimetric Riemannian Penrose Inequality

We prove that the Riemannian Penrose Inequality holds for Asymptotically Flat $3$-manifolds with nonnegative scalar curvature and a connected horizon boundary, provided the optimal decay assumptions are met, which result in the $\mathrm{ADM}$ mass being a well-defined geometric invariant. Our proof builds on new asymptotic comparison arguments involving Huisken's Isoperimetric mass and the Hawking mass, as well as a novel interplay between the Hawking mass and a potential-theoretic version of it, recently introduced by Agostiniani, Oronzio and the third named author. As a crucial step in our argument, we establish a Riemannian Penrose Inequality in terms of the Isoperimetric mass, on any $3$-manifold with nonnegative scalar curvature and connected horizon boundary, on which a well posed notion of weak Inverse Mean Curvature Flow is available. In particular, such an Isoperimetric Riemannian Penrose Inequality does not require the asymptotic flatness of the manifold.Comment: 34 page

The isoperimetric problem on Riemannian manifolds via Gromov-Hausdorff asymptotic analysis

In this paper we prove the existence of isoperimetric regions of any volume in Riemannian manifolds with Ricci bounded below and with a mild assumption at infinity, that is Gromov-Hausdorff asymptoticity to simply connected models of constant sectional curvature. The previous result is a consequence of a general structure theorem for perimeter-minimizing sequences of sets of fixed volume on noncollapsed Riemannian manifolds with a lower bound on the Ricci curvature. We show that, without assuming any further hypotheses on the asymptotic geometry, all the mass and the perimeter lost at infinity, if any, are recovered by at most countably many isoperimetric regions sitting in some Gromov-Hausdorff limits at infinity. The Gromov-Hausdorff asymptotic analysis conducted allows us to provide, in low dimensions, a result of nonexistence of isoperimetric regions in Cartan-Hadamard manifolds that are Gromov-Hausdorff asymptotic to the Euclidean space. While studying the isoperimetric problem in the smooth setting, the nonsmooth geometry naturally emerges, and thus our treatment combines techniques from both the theories.Comment: Minor correction