Abstract
Let
(
M
n
,g)
be a complete Riemannian manifold with
Rc≥−Kg
,
H(x,y,t)
is the heat kernel on
M
n
, and
H=(4πt
)
−
n
2
e
−f
. Nash entropy is defined as
N(H,t)=
∫
M
n
(fH)dμ(x)−
n
2
. We studied the asymptotic behavior of
N(H,t)
and
∂
∂t
[N(H,t)]
as
t→
0
+
, and got the asymptotic formulas at
t=0
. In the Appendix, we got Hamilton-type upper bound for Laplacian of positive solution of the heat equation on such manifolds, which has its own independent interest.