Convergence of the area functional on spaces with lower Ricci bounds and applications

Abstract

We show that the heat flow provides good approximation properties for the area functional on proper $\mathrm{RCD}(K,\infty )$ spaces, implying that in this setting the area formula for functions of bounded variation holds and that the area functional coincides with its relaxation. We then obtain partial regularity and uniqueness results for functions whose hypographs are perimeter minimizing. Finally, we consider sequences of $\mathrm{RCD}(K,N)$ spaces and we show that, thanks to the previously obtained properties, Sobolev minimizers of the area functional in a limit space can be approximated with minimizers along the converging sequence of spaces. Using this last result, we obtain applications on Ricci-limit spaces.