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

TL;DR

This paper demonstrates that heat flow approximates the area functional on $ ext{RCD}(K, ext{infinity})$ spaces, establishing regularity, uniqueness, and convergence results with applications to Ricci-limit spaces.

Contribution

It introduces the use of heat flow to approximate the area functional on $ ext{RCD}(K, ext{infinity})$ spaces and derives regularity, uniqueness, and convergence results.

Findings

Heat flow approximates the area functional effectively.

The area formula for BV functions holds in this setting.

Sobolev minimizers in limit spaces can be approximated by those in converging sequences.

Abstract

We show that the heat flow provides good approximation properties for the area functional on proper $\RCD(K,\infty)$ 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 $\RCD(K,N)$ 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.