Metrics of constant scalar curvatures conformal to a Riemannian product with a round sphere

2008/12/23 by Jimmy Petean, Petean, Jimmy · 1 citation
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations #math.DG

paper · pdf · doi:10.48550/arxiv.0812.4328

9 pages

arxiv created 2008/12/23 · openalex publication_date 2008/12/23 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the conformal class of the Riemannian product g₀ + g, where g₀ 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 Δu -λu + λu^p =0 on S^m, and the number of solutions in terms of λ.

Cited by

• Results on the existence of the Yamabe minimizer of M^m × Rⁿ

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