On the prescribing σ₂ curvature equation on \mathbb S⁴

2009/11/02 by Sun Chang, Zheng-Chao Han, Chang, S. -Y. Alice +3
Mathematics · #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Advanced Mathematical Physics Problems

paper · doi:10.48550/arxiv.0911.0375

Abstract

Prescribing σ_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 σ₂ 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 σ₂ 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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