Consider a Riemannian manifold
(
M
m
,
g
)
(M^{m}, g)
whose volume is the same as the standard sphere
(
S
m
,
g
r
o
u
n
d
)
(S^{m}, g_{round})
. If
p
>
m
2
p\!>\!\frac {m}{2}
and
∫
M
{
R
c
−
(
m
−
1
)
g
}
−
p
d
v
\int _{M}\! \left \{ Rc\!-\!(m\!-\!1)g\right \}_{-}^{p} dv
is sufficiently small, we show that the normalized Ricci flow initiated from
(
M
m
,
g
)
(M^{m}, g)
will exist immortally and converge to the standard sphere. The choice of
p
p
is optimal.