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Petean, Jimmy; Ruiz, Juan Miguel

doi:10.1016/j.difgeo.2013.01.006

Cited by 6 publications

(1 citation statement)

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“…S m × R d for all m ≥ 2, d ≥ 1. We remark that there has been considerable interest in the Yamabe problem on such noncompact manifolds in recent years [1,25,33]. Each of these infinitely many solutions is of the form φ g prod , where g prod is the corresponding product metric g round ⊕ g hyp or g round ⊕ g flat , and φ is a smooth positive function on S m × H d or S m × R d that does not depend on the S m variable, by the asymptotic symmetry method of Caffarelli, Gidas and Spruck [14].…”

Section: Introductionmentioning

confidence: 99%

Infinitely many solutions to the Yamabe problem on noncompact manifolds

Bettiol

Piccione²

2018

Ann. inst. Fourier

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We establish the existence of infinitely many complete metrics with constant scalar curvature on prescribed conformal classes on certain noncompact product manifolds. These include products of closed manifolds with constant positive scalar curvature and simply-connected symmetric spaces of noncompact or Euclidean type; in particular, S m × R d , m ≥ 2, d ≥ 1, and S m × H d , 2 ≤ d < m. As a consequence, we obtain infinitely many periodic solutions to the singular Yamabe problem on S m \ S k , for all 0 ≤ k < (m − 2)/2, the maximal range where nonuniqueness is possible. We also show that all Bieberbach groups in Iso(R d ) are periods of bifurcating branches of solutions to the Yamabe problem on S m × R d , m ≥ 2, d ≥ 1.

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