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