Scalar curvature for manifolds with finite fundamental group
The positive solution of the Yamabe problem shows that
on any closed manifold Mn and in any conformal class
C,
there is a metric g of constant scalar curvature. It is natural
to ask if there is a "best" such metric, and the answer is unfortunately
"no". However, there is a minimal value Y(M, C) of
the scalar curvature of metrics in C with constant scalar curvature
and Riemannian volume 1. The sup Y(M) of
Y(M,
C),
over all conformal classes C, is called the Yamabe invariant of
M. It turns out that
Y(M) > 0 if and only if M
admits a metric of positive scalar curvature, a much studied condition.
(See Stolz's talk in the Spring 2000 meeting.) In dimensions 2, 3, and
4, it is known that Y(M) can be positive, negative, or zero,
and is closely related to geometric structures on
M. In dimensions
5 and up, however, Petean proved that if M is simply connected,
Y(M)
cannot be negative. I will discuss joint work with Boris Botvinnik about
extensions of this result to manifolds with finite fundamental group, and
about the closely related problem of determining what closed manifolds
with finite fundamental group admit metrics of positive scalar curvature.
Open Problems
1. On a closed hyperbolic manifold of dimension bigger than 4, can
one replace the constant curvature metric by another Riemannian metric
with the same volume and constant scalar curvature, so as to INCREASE the
scalar curvature? (This is known to be impossible in low dimensions.
The conjectured answer is no, since at least this cannot be done with a
SMALL change in the metric.)
2. In dimensions bigger than 4, is there some nonlinear elliptic
system (an analogue of Seiberg-Witten) that could be used to produce obstructions
to positive scalar curvature that don't just come from the index theory
of the (linear) Dirac operator? Presumably such a system would, like
Seiberg-Witten, involve a coupling between Dirac and the curvature of some
connection.
3. Given a metric of positive scalar curvature on a sphere Sn
(for which the "index obstruction") vanishes, when can one extend it to
a metric of positive scalar curvature on the ball Dn+1
which is a product metric on a neighborhood of the boundary? This
is basically wide open except when n = 2.
Fall 2000 meeting of the PNGS