On the volume of positively curved Kähler manifolds (Q1104574)

Let M be a complete Kähler manifold of complex dimension n and assume that Ric\(\geq 2(n+1)\delta^ 2\) and \(K_ H\geq 4\delta^ 2\), where Ric and \(K_ H\) denote the Ricci and holomorphic sectional curvature of M, respectively. Here the author's concern is to prove a Bishop-Gromov comparison theorem for M. Let B(p,r) and S(p,r) be the geodesic ball and the geodesic sphere of radius r at \(p\in M\). Then, by computing the mean curvature of S(p,r) with respect to the unit normal of S(p,r), it is proved here that \[ vol(B(p,r))/vol(B(p,r'))\leq V_{\delta,n}(r')/V_{\delta,n}(r),\quad r'\geq r\geq 0, \] where \(V_{\delta,n}(r)\) is the volume of the geodesic ball of radius r in the projective space \(P^ n_{\delta}(C)\). This theorem yields the final result which says that \(vol(M) \leq vol(S_\delta^{2n}\).