We study Riemannian manifolds with boundary under a lower Bakry-Émery Ricci curvature bound. In our weighted setting, we prove several rigidity theorems for such manifolds with boundary. We conclude a rigidity theorem for the inscribed radii, a volume growth rigidity theorem for the metric neighborhoods of the boundaries, and various splitting theorems. We also obtain rigidity theorems for the smallest Dirichlet eigenvalues for the weighted $p$-Laplacians.
"Rigidity of manifolds with boundary under a lower Bakry-Émery Ricci curvature bound." Tohoku Math. J. (2) 71 (1) 69 - 109, 2019. https://doi.org/10.2748/tmj/1552100443