Metrics of constant scalar curvature conformal to Riemannian products

Petean, Jimmy

doi:10.1090/S0002-9939-10-10293-7

Metrics of constant scalar curvature conformal to Riemannian products
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by Jimmy Petean PDF

Proc. Amer. Math. Soc. 138 (2010), 2897-2905 Request permission

Abstract:

We consider the conformal class of the Riemannian product $g_0 +g$, where $g_0$ is the constant curvature metric on $S^m$ and $g$ is a metric of constant scalar curvature on some closed manifold. We show that the number of metrics of constant scalar curvature in the conformal class grows at least linearly with respect to the square root of the scalar curvature of $g$. This is obtained by studying radial solutions of the equation $\Delta u -\lambda u + \lambda u^p =0$ on $S^m$ and the number of solutions in terms of $\lambda$.

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