Monday 17 February 2025, at 11 am in Room 3014 (3rd Floor) - U5 Building, Dr. Enrico Trebeschi will hold the following seminar
Title: Rigidity Results for Maximal Submanifolds in Pseudo-Hyperbolic Space
Abstract: Maximal submanifolds are the generalization in pseudo-Riemannian geometry of minimal surfaces, i.e. submanifolds with identically vanishing mean curvature. They have been studied within the framework pseudo-hyperbolic space, which is the generalization of hyperbolic space in mixed signature, due to 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 upper bound on the norm of the second fundamental form for maximal submanifolds in pseudo-hyperbolic space. In particular, this bound is rigid, meaning it can only be achieved identically, and we are able to explicitly classify maximal submanifolds achieving it. Gauss equation translates the estimate on the extrinsic curvature in terms 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 higher dimension higher rank Teichmüller theory, 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 r > 1.
All interested are invited to participate