Glued spaces and lower Ricci curvature bounds

Abstract

We consider Riemannian manifolds $M_i$, ${i=0,1}$, with boundary and $\Phi_i\in C^{\infty}(M_i)$ non-negative such that the pair $(M_i, \Phi_i)$ admits Bakry-Emery $N$-Ricci curvature bounded from below by $K$. Let $Y_0$ and $Y_1$ be isometric, compact components of the boundary of $M_0$ and $M_1$ respectively and assume $\Phi_0=\Phi_1$ on $Y_0\simeq Y_1$. We assume that $\Pi_0+\Pi_1=\Pi \geq 0$ (*), and $d\Phi_0(\nu_0)+ d\Phi_1(\nu_1)\leq \mbox{tr}\Pi$ on $Y_0\simeq Y_1$ (**) where $\Pi_i$ is the second fundamental form and $\nu_i$ is inner unit normal field along $\partial M_i$. We show that the metric glued space $M=M_0\cup_{\mathcal I}M_1$ together with the measure $\Phi d\mathcal H^n$ satisfies the curvature-dimension condition $CD(K,\lceil N \rceil)$ where $\Phi: M\rightarrow [0,\infty)$ arises tautologically from $\Phi_1$ and $\Phi_2$. Moreover, $(M, \Phi d\mathcal H^n)$ is the collapsed Gromov-Hausdorff limit of smooth, $\lceil N \rceil$-dimensional Riemannian manifolds with Ricci curvature bounded from below by $K- \epsilon$ and is also the measured Gromov-Hausdorff limit of smooth, weighted Riemannian manifolds such that the Bakry-Emery $\lceil N \rceil$-Ricci curvature is bounded from below by $K-\epsilon$. On the other hand we show that given a glued manifold as described it satisfies the curvature-dimension condition $CD(K,N)$ only if the condition (*) and (**) hold. The latter statement generalizes a theorem of Kosovski\u{\i} for sectional lower curvature bounds and especially applies for the unweighted case where a lower Ricci curvature bound and $\dim_{M_i}\leq N$ replaces a lower Bakry-Emery $N$-Ricci curvature bound.