Ricci Curvature and Singularities of Constant Scalar Curvature Metrics

Abstract

In this work we consider periodic spherically symmetric metrics of constant positive scalar curvature on the n-dimensional cylinder called pseudo-cylindric metrics. These metrics belong to the conformal class $[g_0]$ of the Riemannian product $S^1\times S^{n-1}$ : a circle of length $T$ crossed with the (n-1)-dimensional standard sphere. Such metrics have a harmonic Riemannian curvature and a non parallel Ricci tensor, except for the cylindric one. Thus, it appears a natural link between them and the Derdzinski metrics which are warped product and classify a family of Riemannian manifolds. These two families actually differ by conformal transformations. Moreover, we are interested in the multiplicity problem of the pseudo-cylindric metrics in $[g_0]$. We also study the existence problem and the number of the Derdzinski metrics. Furthermore, we prove that the pseudo-cylindric metrics may be expressed in terms of elliptic functions for the dimension $n = 3,4$ and 6 only, and in terms of automorphic functions for any other dimension. This fact allows us to give new bounds for global estimates. Finally, we examine the curvature of the asymptotic pseudo-cylindric metrics which are complete singular Yamabe metrics on the standard sphere punctured of $k$ points. We show that any of such (non trivial) metric has a non parallel Ricci tensor.