  We present multiplicity results for mass constrained Allen-Cahn equations on
a Riemannian manifold with boundary, considering both Neumann and Dirichlet
conditions. These results hold under the assumptions of small mass constraint
and small diffusion parameter. We obtain lower bounds on the number of
solutions according to the Lusternik--Schnirelmann category of the manifold in
case of Dirichlet boundary conditions and of its boundary in the case of
Neumann boundary conditions. Under generic non-degeneracy assumptions on the
solutions, we obtain stronger results based on Morse inequalities. Our approach
combines topological and variational methods with tools from Geometric Measure
Theory.
