A Fully Nonlinear Equation on Four-Manifolds with Positive Scalar Curvature

By M. Gursky, Jeff Viaclovsky. 2003

108 citations recorded by the source corpus.

We present a conformal deformation involving a fully nonlinear equation in dimension 4, starting with positive scalar curvature. Assuming a certain conformal invariant is positive, one may deform from positive scalar curvature to a stronger condition involving the Ricci tensor. We also give a new conformally invariant condition for positivity of the Paneitz operator, which allows us to give many new examples of manifolds admitting metrics with constant $Q$-curvature.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d8ac4128c31aa4b4564c2d13b344ec5ca1db8a4a

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In Applied Mathematics

1. A. Malchiodi
2. Z. Djadli
3. Yanyan Li
4. Jeff Viaclovsky
5. C. B. Ndiaye
6. Sophie Chen
7. M. Gursky
8. Xu-jia Wang

In Geometry and Topology

1. A. Malchiodi
2. Z. Djadli
3. Sophie Chen
4. Jeff Viaclovsky
5. M. Gursky
6. C. B. Ndiaye
7. Yanyan Li
8. L. Martinazzi

Citations in the corpus

• Prescribing symmetric functions of the eigenvalues of the Ricci tensor (2004, background): “8 MATTHEW J. GURSKY AND JEFF A. VIACLOVSKY for solutions of (1.17) (see also [ GV03 ]).”
• Compactness of solutions to some geometric fourth-order equations (2004, background): “Other more general sufficient conditions are given in [32].”
• Existence of conformal metrics with constant Q-curvature (2004, background): “More general conditions for the above hypotheses to hold have been obtained by Gursky and Viaclovsky in [30].”
• Local estimates for some fully nonlinear elliptic equations (2005, background): “The local estimates are derived in [12] and [11] accordingly.”