It is shown that a connected sum of an arbitrary number of complex projective planes carries a metric of positive Ricci curvature with diameter one and, in contrast with the earlier examples of Sha-Yang and Anderson, with volume bounded away from zero. The key step is to construct complete metrics of positive Ricci curvature on the punctured complex projective plane, which have uniform euclidean volume growth and almost contain a line, thus showing topological instability of the splitting theorem of Cheeger—Gromoll, even in the presence of the lower volume bound. In the absence of such a bound, the topological instability was earlier shown by Anderson; metric stability holds, even without the volume bound, by the recent work of Colding-Cheeger.

Outline

We start from a singular space of positive curvature, namely the double spherical suspension of a small round two-sphere. The size of that sphere can be estimated explicitly and is fixed in our construction. The singular points of our space fill a circle of length 2π. We smooth our space near the singular circle in a symmetric way. Then we remove a collection of disjoint small metric balls centered at the former singular points, and glue in our “building blocks” instead. A building block is a metric on ℂP2 \ ball, having positive Ricci curvature and strictly convex boundary. We arrange that the boundary of the building block is isometric to the boundary of the removed ball and is “more convex”. This allows us to smooth the resulting space to get a manifold of positive Ricci curvature.