Given a domain
${\rm\Omega}$
of a complete Riemannian manifold
${\mathcal{M}}$
, define
${\mathcal{A}}$
to be the Laplacian with Neumann boundary condition on
${\rm\Omega}$
. We prove that, under appropriate conditions, the corresponding heat kernel satisfies the Gaussian upper bound
$$\begin{eqnarray}h(t,x,y)\leq \frac{C}{[V_{{\rm\Omega}}(x,\sqrt{t})V_{{\rm\Omega}}(y,\sqrt{t})]^{1/2}}\biggl(1+\frac{d^{2}(x,y)}{4t}\biggr)^{{\it\delta}}e^{-d^{2}(x,y)/4t}\quad \text{for}~t>0,~x,y\in {\rm\Omega}.\end{eqnarray}$$
Here
$d$
is the geodesic distance on
${\mathcal{M}}$
,
$V_{{\rm\Omega}}(x,r)$
is the Riemannian volume of
$B(x,r)\cap {\rm\Omega}$
, where
$B(x,r)$
is the geodesic ball of centre
$x$
and radius
$r$
, and
${\it\delta}$
is a constant related to the doubling property of
${\rm\Omega}$
. As a consequence we obtain analyticity of the semigroup
$e^{-t{\mathcal{A}}}$
on
$L^{p}({\rm\Omega})$
for all
$p\in [1,\infty )$
as well as a spectral multiplier result.