Almost volume cone implies almost metric cone for annuluses centered at a compact set in $RCD(K, N)$-spaces

Abstract

In \cite{CC1}, Cheeger-Colding considered manifolds with lower Ricci curvature bound and gave some almost rigidity results about warped products including almost metric cone rigidity and quantitative splitting theorem. As a generalization of manifolds with lower Ricci curvature bound, for metric measure spaces in $RCD(K, N)$, $1<N<\infty$, splitting theorem \cite{Gi13} and "volume cone implies metric cone" rigidity for balls and annuluses of a point \cite{PG} have been proved. In this paper we will generalize Cheeger-Colding's \cite{CC1} result about "almost volume cone implies almost metric cone for annuluses of a compact subset " to $RCD(K, N)$-spaces. More precisely, consider a $RCD(K, N)$-space $(X, d, \mathfrak m)$ and a Borel subset $\Omega\subset X$. If the closed subset $S=\partial \Omega$ has finite outer curvature, the diameter ${diam}(S)\leq D$ and the mean curvature of $S$ satisfies $$m(x)\leq m, \, \forall x\in S,$$ and \begin{equation*}\mathfrak m(A_{a, b}(S))\geq (1-\epsilon)\int_a^b \left({sn}'_H(r)+ \frac{m}{n-1}{sn}_H(r)\right)^{n-1}dr \mathfrak m_S(S)\end{equation*} then $A_{a', b'}(S)$ is measured Gromov-Hausdorff close to a warped product $(a', b')\times_{{sn}'_H(r)+ \frac{m}{n-1}{sn}_H(r)}Y,$ $A_{a, b}(S)=\{x\in X\setminus \Omega, \, a<d(x, S)<b\}$, $a<a'<b'<b$, $Y$ is a metric space with finite components with each component is a $RCD(0, N-1)$-space when $m=0, K=0$ and is a $RCD(N-2, N-1)$-space for other cases and $H=\frac{K}{N-1}$. Note that when $m=0, K=0$, our result is a kind of quantitative splitting theorem and in other cases it is an almost metric cone rigidity. To prove this result, different from \cite{Gi13, PG}, we will use \cite{GiT}'s second order differentiation formula and a method similar as \cite{CC1}.