2023/08/13 by Ketterer, Christian
#53C21 #54E35 #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2308.06848
We consider Riemannian manifolds Mi, i=0,1, with boundary and Φi∈ C∞(Mi) non-negative such that the pair (Mi, Φi) admits Bakry-Emery N-Ricci curvature bounded from below by K. Let Y0 and Y1 be isometric, compact components of the boundary of M0 and M1 respectively and assume Φ0=Φ1 on Y0≃ Y1. We assume that Π0+Π1=Π≥ 0 (*), and dΦ0(ν0)+ dΦ1(ν1)≤ trΠ on Y0≃ Y1 (**) where Πi is the second fundamental form and νi is inner unit normal field along ∂ Mi. We show that the metric glued space M=M0∪\mathcal IM1 together with the measure Φd\mathcal Hn satisfies the curvature-dimension condition CD(K,\lceil N \rceil) where Φ: M→ [0,∞) arises tautologically from Φ1 and Φ2. Moreover, (M, Φd\mathcal Hn) 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 dimMi≤ N replaces a lower Bakry-Emery N-Ricci curvature bound.