The scalar curvature problem on four-dimensional manifolds

Hichem Chtioui; Hichem Hajaiej; Marwa Soula

doi:10.3934/cpaa.2020034

Article Contents

2020, Volume 19, Issue 2: 723-746. Doi: 10.3934/cpaa.2020034

This issue Previous Article Prescribing

$ Q $

-curvature on

$ S^n $

in the presence of symmetry Next Article Decay of solutions for a dissipative higher-order Boussinesq system on a periodic domain

The scalar curvature problem on four-dimensional manifolds

1.
Department of mathematics, Faculty of Sciences of Sfax, 3018 Sfax, Tunisia
2.
California State University Los Angeles, 5151 University Drive, Los Angeles, Department of mathematics, Faculty of Sciences of Sfax, 3018 Sfax, Tunisia

Received: September 30, 2018

Revised: June 30, 2019

Published: October 2019

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Abstract

We study the problem of existence of conformal metrics with prescribed scalar curvatures on a closed Riemannian $ 4 $-manifold not conformally diffeomorphic to the standard sphere $ S^{4} $. Using the critical points at infinity theory of A.Bahri [6] and the positive mass theorem of R.Schoen and S.T.Yau [32], we prove compactness and existence results under the assumption that the prescribed function is flat near its critical points. These are the first results on the prescribed scalar curvature problem where no upper-bound condition on the flatness order is assumed.

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Mathematics Subject Classification: Primary: 35J60; Secondary: 58E05.

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References

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