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On the prescribing σ 2 curvature equation on \({\mathbb S^4}\)

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Abstract

Prescribing σ k 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 σ 2 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 σ 2 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 σ 2 curvature equation with the given K using a degree argument involving fully nonlinear elliptic operators to the above deformation.

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Authors and Affiliations

  1. Department of Mathematics, Princeton University, Princeton, NJ, 08540, USA

    Sun-Yung Alice Chang & Paul Yang

  2. Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ, 08854, USA

    Zheng-Chao Han

Authors
  1. Sun-Yung Alice Chang
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  2. Zheng-Chao Han
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  3. Paul Yang
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Correspondence to Zheng-Chao Han.

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Communicated by N. Trudinger.

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Chang, SY.A., Han, ZC. & Yang, P. On the prescribing σ 2 curvature equation on \({\mathbb S^4}\) . Calc. Var. 40, 539–565 (2011). https://doi.org/10.1007/s00526-010-0350-2

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  • DOI: https://doi.org/10.1007/s00526-010-0350-2

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