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Invited talk

Wednesday, July 21, 14:45 ~ 15:45 UTC-3

On the stability of homogeneous Einstein manifolds

Jorge Lauret

Universidad Nacional de Córdoba and CIEM (CONICET), Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $M$ be a compact differentiable manifold and let $\mathcal{M}_1$ denote the space of all unit volume Riemannian metrics on $M$.  The critical points of the simplest curvature functional, given by the total scalar curvature $ Sc:\mathcal{M}_1\rightarrow {\Bbb R}, $ are called Einstein metrics and play a fundamental role in Differential Geometry and Physics.  Among Einstein metrics with positive scalar curvature, those which are stable as critical points of Sc (i.e., negative definite Hessian) on the subspace $\mathcal{C}_1$ of unit volume constant scalar curvature metrics on $M$, and in particular local maxima of $Sc|_{\mathcal{C}_1}$, seems to be extremely rare.      

In this talk, after some general preliminaries, we will focus on the case when the metrics and the variations are considered to be $G$-invariant for some compact Lie group $G$ acting transitively on $M$.  This takes us to work on the overwhelming and sophisticated class of all compact homogeneous spaces.  

Joint work with Emilio Lauret (Universidad Nacional del Sur and INMABB (CONICET), Argentina) and Cynthia Will (Universidad Nacional de Córdoba and CIEM (CONICET), Argentina).

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