Glued spaces and lower Ricci curvature bounds

Abstract

We consider Riemannian manifolds $M_i$, ${i=0,1}$, with boundary and $Φ_i\in C^{\infty}(M_i)$ non-negative such that the pair $(M_i, Φ_i)$ 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 $Φ_0=Φ_1$ on $Y_0\simeq Y_1$. We assume that $Π_0+Π_1=Π\geq 0$ (*), and $dΦ_0(ν_0)+ dΦ_1(ν_1)\leq \mbox{tr}Π$ on $Y_0\simeq Y_1$ (**) where $Π_i$ is the second fundamental form and $ν_i$ 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 $Φd\mathcal H^n$ satisfies the curvature-dimension condition $CD(K,\lceil N \rceil)$ where $Φ: M\rightarrow [0,\infty)$ arises tautologically from $Φ_1$ and $Φ_2$. Moreover, $(M, Φd\mathcal H^n)$ is the collapsed Gromov-Hausdorff limit of smooth, $\lceil N \rceil$-dimensional Riemannian manifolds with Ricci curvature bounded from below by $K- ε$ 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 $K-ε$. 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 $\dim_{M_i}\leq N$ replaces a lower Bakry-Emery $N$-Ricci curvature bound.