On measure contraction property without Ricci curvature lower bound

TL;DR

This paper explores measure contraction properties in non-smooth metric measure spaces, extending known results to certain weakly Sasakian manifolds that satisfy MCP conditions without Ricci curvature bounds.

Contribution

It provides sufficient conditions for weakly Sasakian manifolds to satisfy MCP(0,2n+3), broadening understanding beyond classical Ricci curvature bounds.

Findings

Weakly Sasakian manifolds satisfy MCP(0,2n+3) under certain conditions

Extension of MCP results from Heisenberg group to broader class of manifolds

Demonstrates MCP can hold without Ricci curvature bounded below

Abstract

Measure contraction properties $MCP(K,N)$ are synthetic Ricci curvature lower bounds for metric measure spaces which do not necessarily have smooth structures. It is known that if a Riemannian manifold has dimension $N$, then $MCP(K,N)$ is equivalent to Ricci curvature bounded below by $K$. On the other hand, it was observed in \cite{Ri} that there is a family of left invariant metrics on the three dimensional Heisenberg group for which the Ricci curvature is not bounded below. Though this family of metric spaces equipped with the Harr measure satisfy $MCP(0,5)$. In this paper, we give sufficient conditions for a $2n+1$ dimensional weakly Sasakian manifold to satisfy $MCP(0,2n+3)$. This extends the above mentioned result on the Heisenberg group in \cite{Ri}.