Allen-Cahn Approximation of Mean Curvature Flow in Riemannian manifolds I, uniform estimates

We are concerned with solutions to the parabolic Allen-Cahn equation in Riemannian manifolds. For a general class of initial condition we show non positivity of the limiting energy discrepancy. This in turn allows to prove almost monotonicity formula (a weak counterpart of Huisken's monotonicity formula) which gives a local uniform control of the energy densities at small scales. Such results will be used in [40] to extend previous important results from [31] in Euclidean space, showing convergence of solutions to the parabolic Allen-Cahn equations to Brakke's motion by mean curvature in space forms.