Buongiorno,
Ricordo a tutti gli interessati che il prossimo (e ultimo per questo semestre) seminario di geometria sarà tenuto da Enrico Trebeschi e si terrà Martedì 10 Dicembre alle ore 14:30 presso l'aula Riunioni.
Per gli interessati ad andare a pranzo con lo speaker, l'appuntamento è in atrio alle 13:00.
Allego titolo e abstract:
Titolo: Rigidity Results for Maximal Submanifolds in Pseudo-Hyperbolic Space
Abstact: Maximal submanifolds are the generalization in pseudo-Riemannian geometry of minimal surfaces. Their study in pseudo-hyperbolic space, which is the generalization of hyperbolic space in mixed signature, is motivated by several factors ranging from differential geometry, relativistic physics, and, in recent years, geometric topology.
A classical problem in the study of minimal submanifolds of the unit sphere is to find a universal upper bound for the norm of the second fundamental form, that is, to estimate their extrinsic curvature.
In this seminar, I will discuss a recent work in collaboration with Alex Moriani (Université Côte d'Azur), in which we provide a sharp bound on the norm of the second fundamental form for maximal submanifolds in pseudo-hyperbolic space. In particular, this bound is rigid, meaning it is achieved identically, and we are able to explicitly classify the submanifolds that achieve it.
Gauss equation translates the estimate on the extrinsic curvature in trems of the intrinsic curvature, allowing us to prove that the scalar curvature of a maximal submanifold is non-positive and to classify those with vanishing scalar curvature. In the Lorentzian case, we are able to provide a rigid estimate on the Ricci curvature as well.
If I have time, I would like to discuss the implications of this work in the context of Teichmüller theory of higher rank and dimension, namely the study of connected components of Hom(π1(M), G) containing only faithful and discrete representations, where M is a closed manifold of dimension n>2 and G is a semi-simple Lie group of rank >1.
A presto,
Carlo e Filippo