Decay of solutions for a dissipative higher-order Boussinesq system on a periodic domain
Prescribing $ Q $-curvature on $ S^n $ in the presence of symmetry
February 2020, 19(2): 723-746. doi: 10.3934/cpaa.2020034
The scalar curvature problem on four-dimensional manifolds
Hichem Chtioui 1, , Hichem Hajaiej 2, and Marwa Soula 2,
Department of mathematics, Faculty of Sciences of Sfax, 3018 Sfax, Tunisia
California State University Los Angeles, 5151 University Drive, Los Angeles, Department of mathematics, Faculty of Sciences of Sfax, 3018 Sfax, Tunisia
Received October 2018 Revised July 2019 Published October 2019
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.
Keywords: Conformal metrics, scalar curvature, variational method, loss of compactness, critical points at infinity.
Mathematics Subject Classification: Primary: 35J60; Secondary: 58E05.
Citation: Hichem Chtioui, Hichem Hajaiej, Marwa Soula. The scalar curvature problem on four-dimensional manifolds. Communications on Pure & Applied Analysis, 2020, 19 (2) : 723-746. doi: 10.3934/cpaa.2020034
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Hichem Chtioui Hichem Hajaiej Marwa Soula