Einstein metrics and volume entropy

This talk will explain a paper written jointly with M. Brunnbauer and M. Ishida,
(http://www.mrlonline.org/mrl/2009-016-003/2009-016-003-010.pdf [4]) .

The volume entropy of a smooth metric is the entropy of the volume of a ball in the
universal covering of a the Riemannian manifold. The  minimal volume is the infimum
taken over all unit volume metrics.

We showed (as a consecuence of Brunnbauer's deep results) that  performing a
connected sum with a manifold that is not essential (in Gromov's terminology) does
not affect the minimal volume entropy.

Using this, we computed estimates for the minimal volume entropy of  certain families
of smooth 4-manifolds and used the Bauer-Furuta invariants (as a consequence of
results of Ishida, Lebrun and Sasahira) to present an infinite family of 4-manifolds with the
following properties:
(i)They have positive minimal volume entropy.
(ii)They satisfy a strict version of the Gromov-Hitchin-Thorpe
inequality, with a minimal volume entropy term.
(iii)They nevertheless admit infinitely many distinct smooth structures
for which no compatible Einstein metric exists.

This answers a question LeBrun posed in 1994.

 The first half will focus on the notions of volume entropy, Gromov's definition
of "essential manifold", what the Bauer-Furuta invariants allow us to do and
their relation to the Seiberg-Witten invariants.

In the second half (ideally) we should see sketches of the first  results regarding
the volume entropy and the numerology involved in  getting all the topological
invariants to obey (ii).