Convergence of the area functional on spaces with lower Ricci bounds and applications
Alessandro Cucinotta
TL;DR
The paper develops a robust bridge between smooth and non-smooth geometric analysis under lower Ricci curvature bounds by studying the area functional on $CD(K,N)$ spaces and their limits. It shows that heat flow provides precise approximation of the area functional on proper $CD(K, infty)$ spaces, yielding an area formula that equates area with the perimeter of hypographs and proving that the relaxed area equals the area. It further obtains partial regularity and uniqueness results for hypographs of perimeter-minimizing functions, and proves a key approximation theorem allowing Sobolev minimizers of the area on limit spaces to be approximated by minimizers along converging sequences of spaces, enabling applications to Ricci limit spaces. Collectively, these results equip the non-smooth Ricci-curvature setting with tools to transfer minimizers and regularity from smooth manifolds to their limits, and to analyze convergence under pmGH for sequences of $CD(K,N)$ spaces. The findings have implications for the study of minimal graphs and perimeter-minimizing structures in Ricci-limit contexts, with potential reach into geometric measure theory on non-smooth spaces.
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.