Comparison geometry of manifolds with boundary under a lower weighted   Ricci curvature bound
  We study Riemannian manifolds with boundary under a lower weighted Ricci curvature bound. We consider a curvature condition in which the weighted Ricci curvature is bounded from below by the density function. Under the curvature condition, and a suitable condition for the weighted mean curvature for the boundary, we obtain various comparison geometric results. 
Yohei Sakurai
math.DG