On the prescribing $σ_2$ curvature equation on $\mathbb S^4$

Details On the prescribing $σ_2$ curvature equation on $\mathbb S^4$ On the prescribing $σ_2$ curvature equation on $\mathbb S^4$

2009-11-02

arxiv.org/abs/0911.0375v2

Mathematics

Differential Geometry

39 pages. Statement and proof of Corollary 1 have been revised. A remark added on p. 34, and a typo corrected two lines below

Scientific paper

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 $K$.

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