 
Summary: RESEARCH BLOG 7/23/03
Let (M, g) be a Riemannian manifold. Recall that we defined
µ(g, ) = inf{
M
[( u2
+ R) + u]eu
dV  n 
n
2
ln(4)

M
eu
dV = 1}.
(see research blog 7/21/03 ). We will use V k
n (r) to denote the volume
of a ball of radius r in the simplyconnected space form of sectional
curvature k of dimension n, and Br(x) = Br(x, g) denotes the ball of
radius r in M centered at the point x in the metric g.
Lemma 0.1. Suppose that R(g) ^R, and Ric(g) (n  1)kg, then
