\section{Introduction}
Monolayer helium adsorbed on graphite has long proven to be a 
fascinating system for studying the growth and behavior of a quantum
film above an atomically ordered, uniform substrate and has
been used to investigate a number of nearly two-dimensional (2D) 
phenomena. \cite{dash75,schick80,bruch97}
A very prominent feature of this layer is the
$\sqrt 3 \times \sqrt3$ commensurate solid phase in which 
one-third of the available 
substrate adsorption sites are occupied.
\cite{bretz77,greywall93,nielsen80,abraham87}
For densities above the commensurate, the system passes
through a region of domain wall phases before forming an
incommensurate triangular solid phase.\cite{carneiro76,hering76,rabedeau89}
The nature
of the phase diagram below the commensurate density at low 
temperatures is less well
established, with two competing pictures.  One possibility is that
the phase is a solid with clustered vacancies. \cite{ecke85}  
More recently, it has been suggested that the solid melts
if the density is decreased, forming a 
low temperature liquid phase \cite{greywall91} that 
could be a candidate in the search for a monolayer
superfluid with no underlying inert 
layer.\cite{greywall91,chen94,boninsegni98,cowan98}

The solid cluster picture has been 
discussed by Ecke {\em et al}. \cite{ecke85}
They note that since the commensurate solid
phase is in the same universality class as the three-state 
Potts model, \cite{bretz73,berker78,schick77,tejwani80,crary87} then at 
lower densities the film should consist of a commensurate solid
with vacancies.  If the temperature is raised, the solid 
melts continuously.  Lowering the temperature causes the vacancies to
coalesce (phase separate), a first order transition.  The difference 
between the temperatures of these two transitions, signaled
by heat capacity peaks, becomes smaller
as the density is lowered until they meet at a tricritical point, which
occurs at about
0.039 atom/$\AA^2$ and 1.3 K. \cite{ecke85}  Thus, in this picture,
the monolayer consists of solid clusters and a low density vapor
at low temperatures and densities.

More recent experiments have questioned this conclusion.  Greywall
and Busch \cite{greywall91} point out that the heat capacity is not 
linear in density for the entire region below the commensurate density, 
as it must be for phase coexistence.  They instead
propose that a self-bound liquid phase occurs at about 0.04
atom/$\AA^2$.  This conclusion is supported by 2D variational 
calculations. \cite{bruch93} 
However, direct 
measurements \cite{reppy94,reppy96,mohandas95} detect 
no superfluidity, possibly because of poor substrate connectivity.

In a previous publication\cite{pierce98}, we have used 
path-integral Monte Carlo method and realistic helium-helium and 
helium graphite interactions to study the monolayer at or below 
the commensurate density. We found that the presence of corrugations,
which causes the commensurate solid at 1/3 coverage, creates
solid commensurate clusters at densities below the commensurate 
density. We found that it is the melting of these monolayer 
clusters which gives rise to a specific heat maximum which was incorrectly 
interpreted as onset of monolayer superfluidity. In this paper we examine in detail 
the role of substrate corrugation on the first layer phase diagram 
using the path-integral Monte Carlo method.  
First, we present results for helium adsorbed on a smooth 
substrate.  The helium-graphite interaction is based on the laterally
averaged potential of Carlos and Cole.\cite{cole80}
This layer exhibits
liquid and solid phases, and we calculate the phase diagram at low
temperatures. These results are compared with recent Green's function Monte Carlo 
calculations.\cite{whitlock98}
The second layer promotion density is also determined.  
Next, calculations including substrate
corrugations are discussed, and direct comparisons of the melting
behavior, specific heat, and single particle binding energy that
we obtain are made
with experiment.  Finally, having verified our method by the above
comparisons, we examine the low density, low temperature phase of the
helium monolayer.  We determine that for all coverages below the 
commensurate density, the system consists of solid clusters in
phase coexistence with coalesced vacancies.  No liquid phase
occurs and so there is no possibility for first layer superfluidity.

