Tristan Ozuch-Meersseman (ENS Paris, France) will speak in the geometry seminar on Tuesday 19 March, at 1.30pm in the room OF 2070. Tristan’s title is Noncollapsed degeneration of Einstein 4-manifolds and his abstract is below.
A theorem by Anderson and Bando-Kasue-Nakajima from 1989 states that in
dimension 4, to compactify the set of noncollapsed Einstein metrics with
an upper bound on the diameter in the Gromov-Hausdorff (GH) sense, one has
to add singular spaces called Einstein orbifolds and the singularities
form as blow-downs of Ricci-flat ALE spaces.
This raises some natural issues formulated by Anderson. Which Einstein
orbifolds can be expressed as GH limits of Einstein manifolds ? Does this
completion of the moduli space of Einstein metrics admit a smooth
structure ?
We partially answer these by first proving that every Einstein manifold
sufficiently GH-close to an Einstein orbifold is the result of a
glueing-perturbation procedure, which is a generalization of Biquard’s
construction. This enables us to describe any noncollapsed degeneration of
compact Einstein metrics and to exhibit obstructions to approximate
Einstein orbifolds by smooth Einstein metrics in the GH sense.