 
Summary: ON UNIQUENESS AND DIFFERENTIABILITY IN THE SPACE OF YAMABE
METRICS
MICHAEL T. ANDERSON
Abstract. It is shown that there is a unique Yamabe representative for a generic set of conformal
classes in the space of metrics on any manifold. At such classes, the scalar curvature functional is
shown to be dierentiable on the space of Yamabe metrics. In addition, some suÆcient conditions are
given which imply that a Yamabe metric of locally maximal scalar curvature is necessarily Einstein.
1. Introduction.
Let M be a closed ndimensional manifold. For a given smooth metric g on M , let [g] denote the
conformal class of g, consisting of smooth metrics on M pointwise conformal to g. By the solution to
the Yamabe problem [1], [6], in each conformal class [g] there is a Yamabe metric
of constant scalar
curvature s
; the metric
minimizes the total scalar curvature (or EinsteinHilbert action)
S(g) = v (n 2)=n
Z
M
s g dV g :
when S is restricted to the class of conformal metrics
g 2 [g]. Here s g denotes the scalar curvature of
g, dV g the volume form, and v the total volume of (M; g). The sign of s
, i.e s
< 0, s
= 0 or s
> 0
depends only on the conformal class [
].
