  The goal of this note is to prove convergence results w.r.t. the area
functional on metric measure spaces with a lower Ricci curvature bound. The
paper can be divided in two parts. In the first part, we show that the heat
flow provides good approximation properties for the area functional on
$RCD(K,\infty)$ spaces, implying that in this setting the area formula for
functions of bounded variation holds and that the area functional coincides
with its relaxation. Moreover, we obtain partial regularity and uniqueness
results for functions whose epigraphs are perimeter minimizing. In the second
part of the paper, we consider Ricci limit spaces (and also finite dimensional
$RCD(K,N)$ spaces for some results) and we show that, thanks to the previously
obtained properties, minimizers of the area functional can be approximated with
minimizers along the converging sequence of manifolds. As a first application,
we show that minimizers of the area functional on non-collapsed Ricci limit
spaces are locally Lipschitz and satisfy a-priori gradient estimates. Secondly,
we obtain a Bernstein-type result for area minimizers with sublinear growth at
infinity.
