METRIC RIGIDITY OF KAHLER MANIFOLDS WITH LOWER RICCI BOUNDS AND ALMOST MAXIMAL VOLUME

In this short note we prove that a Kahler manifold with lower Ricci curvature bound and almost maximal volume is Gromov-Hausdorff close to the projective space with the Fubini-Study metric. This is done by combining the recent results on holomorphic rigidity of such Kahler manifolds (see Gang Liu [Asian J. Math. 18 (2014), 69-99]) with the structure theorem of Tian-Wang (see Gang Tian and Bing Wang [J. Amer. Math. Soc 28 (2015), 1169-1209]) for almost Einstein manifolds. This can be regarded as the complex analog of the result on Colding on the shape of Riemannian manifolds with almost maximal volume.