On a compact Riemannian manifold with boundary, we study how Ricci curvature of the interior affects the geometry of the boundary. First, we establish integral inequalities for functions defined solely on the boundary and apply them to obtain geometric inequalities involving the total mean curvature. Then, we discuss related rigidity questions and prove Ricci curvature rigidity results for manifolds with boundary.
"Boundary effect of Ricci curvature." J. Differential Geom. 103 (1) 59 - 82, May 2016. https://doi.org/10.4310/jdg/1460463563