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On the prescribing $\sigma_2$ curvature equation on $\mathbb S^4$
- Publication Year :
- 2009
-
Abstract
- Prescribing $\sigma_k$ curvature equations are fully nonlinear generalizations of the prescribing Gaussian or scalar curvature equations. Given a positive function $K$ to be prescribed on the 4-dimensional round sphere. We obtain asymptotic profile analysis for potentially blowing up solutions to the $\sigma_2$ curvature equation with the given $K$; and rule out the possibility of blowing up solutions when $K$ satisfies a non-degeneracy condition. We also prove uniform a priori estimates for solutions to a family of $\sigma_2$ curvature equations deforming $K$ to a positive constant under the same non-degeneracy condition on $K$, and prove the existence of a solution using degree argument to this deformation involving fully nonlinear elliptic operators under an additional, natural degree condition on a finite dimensional map associated with $K$.<br />Comment: 39 pages. Statement and proof of Corollary 1 have been revised. A remark added on p. 34, and a typo corrected two lines below equation (68) on p.32
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.0911.0375
- Document Type :
- Working Paper