| Abstract |
Einstein metrics can be characterised as critical points of the (normalised)
total scalar curvature functional. They are always saddle points. However,
there are Einstein metrics which are local maxima of the functional restricted
to metrics of fixed volume and constant scalar curvature. These are by
definition stable Einstein metrics. Stability can equivalently be characterised
by a spectral condition for the Lichnerowicz Laplacian on divergence-and
trace-free symmetric 2-tensors, i.e. on so-called tt-tensors:an Einstein
metric is stable if twice the Einstein constant is a lower bound for this
operator.
In my talk I want to discuss the stability condition. I will present a
results obtained with G. Weingart, completing the work of Koiso on the
classification of stable compact symmetric spaces. Moreover, I will describe
an interesting relation between instability and the existence of harmonic
forms. This is done in the case of nearly K\"ahler, Einstein-Sasaki
and nearly G_2 manifolds. If time permits I will also explain the instability
of the Berger space SO(5)/SO(3), which is a homology sphere. In this case
instability surprisingly is related to the existence of Killing tensors.
The latter results are contained in joint work with M. Wang and C. Wang. |
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