  For a given bounded domain $\Omega$ with smooth boundary in a smooth
Riemannian manifold $(\mathcal{M},g)$, we show that the Poisson type
upper-estimate of the heat kernel associated to the Dirichlet-to-Neumann
operator, the Sobolev trace inequality, the Log-Sobolev trace inequality, the
Nash trace inequality, and the Rozenblum-Lieb-Cwikel type inequality are all
equivalent. Upon decomposing the Dirichlet-to-Neumann operator into a sum of
the square root of the Laplacian and a pseudodifferntial operator and by
applying Grubb's method of symbolic calculus for the corresponding
pseudodifferential heat kernel operators, we establish a procedure to calculate
all the coefficients of the asymptotic expansion of the trace of the heat
kernel associated to Dirichlet-to-Neumann operator as $t\to 0^+$. In
particular, we explicitly give the first four coefficients of this asymptotic
expansion. These coefficients give precise information regarding the area and
curvatures of the boundary of the domain in terms of the spectrum of the
Steklov problem.
