Asymptotic isoperimetry on non collapsed spaces with lower Ricci bounds
Gioacchino Antonelli, Enrico Pasqualetto, Marco Pozzetta, Daniele Semola
Abstract
Abstract This paper studies sharp and rigid isoperimetric comparison theorems and asymptotic isoperimetric properties for small and large volumes on N -dimensional $$\textrm{RCD}(K,N)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>RCD</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> spaces $$(X,\textsf {d},\mathscr {H}^N)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>N</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . Moreover, we obtain almost regularity theorems formulated in terms of the isoperimetric profile and enhanced consequences at the level of several functional inequalities. Most of our statements seem to be new even in the classical setting of smooth, non compact manifolds with lower Ricci curvature bounds. The synthetic theory plays a key role via compactness and stability arguments.