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 ${\Pi }_0+{\Pi }_1=\Pi \ge 0$ (*), and $dΦ_0(ν_0)+ dΦ_1(ν_1)\leq \mbox{tr}Π$ 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 $\Phi :M\to [0,\infty )$ 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\uı for sectional lower curvature bounds and especially applies for the unweighted case where a lower Ricci curvature bound and ${\dim }_{M_i}\le N$ replaces a lower Bakry-Emery $N$-Ricci curvature bound.

44 pages, final version