On Kähler manifolds with Ricci curvature bounded from below, we establish some theorems which are counterparts of some classical theorems in Riemannian geometry, for example, Bishop-Gromov’s relative volume comparison, Bonnet-Meyers theorem, and Yau’s gradient estimate for positive harmonic functions. The tool is a Bochner type formula reflecting the Kähler structure.
"Kähler manifolds with Ricci curvature lower bond." Asian J. Math. 18 (1) 69 - 100, January 2014.