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ON UNIQUENESS AND DIFFERENTIABILITY IN THE SPACE OF YAMABE MICHAEL T. ANDERSON
 

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 di erentiable on the space of Yamabe metrics. In addition, some suĂcient conditions are
given which imply that a Yamabe metric of locally maximal scalar curvature is necessarily Einstein.
1. Introduction.
Let M be a closed n-dimensional 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 Einstein-Hilbert 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 [ ].

  

Source: Anderson, Michael - Department of Mathematics, SUNY at Stony Brook

 

Collections: Mathematics