On the prescribing σ2 curvature equation on S4

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Abstract

Prescribing σk curvature equations are fully nonlinear generalizations of the prescribing Gaussian or scalar curvature equations. For a 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. Under the same non-degeneracy condition on K, we also prove uniform a priori estimates for solutions to a family of σ2 curvature equations deforming K to a positive constant; and under an additional, natural degree condition on a finite dimensional map associated with K, we prove the existence of a solution to the σ2 curvature equation with the given K using a degree argument involving fully nonlinear elliptic operators to the above deformation.

Original languageEnglish (US)
Pages (from-to)539-565
Number of pages27
JournalCalculus of Variations and Partial Differential Equations
Volume40
Issue number3
DOIs
StatePublished - Jan 24 2011

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

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