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## Rigidity and stability of Einstein metrics on homogeneous spaces

### Paul Schwahn

#### University of Stuttgart, Germany   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak56c1e878e34b1a41b0713fee03768f45').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy56c1e878e34b1a41b0713fee03768f45 = 'p&#97;&#117;l.schw&#97;hn' + '&#64;'; addy56c1e878e34b1a41b0713fee03768f45 = addy56c1e878e34b1a41b0713fee03768f45 + 'm&#97;th&#101;m&#97;t&#105;k' + '&#46;' + '&#117;n&#105;-st&#117;ttg&#97;rt' + '&#46;' + 'd&#101;'; var addy_text56c1e878e34b1a41b0713fee03768f45 = 'p&#97;&#117;l.schw&#97;hn' + '&#64;' + 'm&#97;th&#101;m&#97;t&#105;k' + '&#46;' + '&#117;n&#105;-st&#117;ttg&#97;rt' + '&#46;' + 'd&#101;';document.getElementById('cloak56c1e878e34b1a41b0713fee03768f45').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy56c1e878e34b1a41b0713fee03768f45 + '\'>'+addy_text56c1e878e34b1a41b0713fee03768f45+'<\/a>';

The question of rigidity of a given Einstein metric $g$, i.e. whether $g$ can be deformed through a curve $g_t$ of Einstein metrics on the same manifold, is closely related to the stability of $g$ under the Einstein-Hilbert action by the fact that Einstein metrics are critical points of the (normalized) total scalar curvature functional. The stability problem for irreducible compact symmetric spaces has been widely investigated by N. Koiso and settled by recent results, using the theory of harmonic analysis on homogeneous spaces.

I give an overview of the results about rigidity and stability on symmetric spaces and present some novel results about the rigidity and stability of the non-symmetric homogeneous spaces like, for example, the 6-dimensional homogeneous nearly Kähler manifolds.

Joint work with Uwe Semmelmann (University of Stuttgart, Germany).