  This paper investigates the question of which smooth compact 4-manifolds
admit Riemannian metrics that minimize the L2-norm of the curvature tensor.
Metrics with this property are called OPTIMAL; Einstein metrics and scalar-flat
anti-self-dual metrics provide us with two interesting classes of examples.
Using twistor methods, optimal metrics of the second type are constructed on
the connected sums kCP_2 for k > 5. However, related constructions also show
that large classes of simply connected 4-manifolds do not admit any optimal
metrics at all. Interestingly, the difference between existence and
non-existence turns out to delicately depend on one's choice of smooth
structure; there are smooth 4-manifolds which carry optimal metrics, but which
are homeomorphic to infinitely many distinct smooth 4-manifolds on which no
optimal metric exists.
