Pyramids in the complex projective plane |
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Authors: | Mikhail Katz |
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Affiliation: | (1) Département de Mathématiques, Université de Nancy 1, B.P. 239, 54506 Vandoeuvre, France |
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Abstract: | We study the critical points of the diameter functional 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, Pk CP1 CP2, k=1,2... We show that Pk is a local minimum of by verifying the positivity of the Hessian of (a smooth approximation to) at Pk. 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 , given by a subset which is not contained in any totally geodesic submanifold of CP2. |
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