# Stability of Einstein metrics

##
Uwe Semmelmann

created by daniele on 26 Jan 2021

modified on 23 Feb 2021

25 feb 2021
-- 14:30

Firenze (online)

Seminario di Geometria Differenziale e Analisi Complessa del Dipartimento di Matematica e Informatica "Ulisse Dini" dell'Università di Firenze

**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. Stability is also related to Perelman's
\nu entropy and dynamical stability with respect to the Ricci flow.

In my talk I will discuss the stability condition. I will present a
recent result obtained with G. Weingart, which completes 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
Kähler, 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.