Pyramids in the complex projective plane

We study the critical points of the diameter functional δ on the n-fold Cartesian product of the complex projective plane CP² with the Fubini-Study metric. Such critical points arise in the calculation of a metric invariant called the filling radius, and are akin to the critical points of the distance function. We study a special family of such critical points, P_k⊂CP¹⊂CP², k=1,2... We show that P_k is a local minimum of δ by verifying the positivity of the Hessian of (a smooth approximation to) δ at P_k. For this purpose, we use Shirokov's law of cosines and the holonomy of the normal bundle of CP¹⊂CP². We also exhibit a critical point of δ, given by a subset which is not contained in any totally geodesic submanifold of CP².