Ngô Quốc Anh

August 29, 2014

Prescribed Q-curvature and scalar curvature problems in the null case

Filed under: PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 20:24

On a 2-dimensional compact Riemannian manifold (M, g) without boundary, the prescribed scalar curvature problem in the flat case is equivalent to solving the following PDE

\displaystyle -\Delta_g u = fe^{2u}

with f is a given non-constant smooth function on M and \Delta_g is the Laplace-Beltrami operator associated with the metric g.

Simply by integrating both sides of the PDE, it is immediate to see that if u solves the PDE, it would satisfy \int_M f e^{2u} dv =0; hence the candidate function f must change sign in M. In their elegant paper published in 1974, Kazdan and Warner showed that in addition to the sign-changing property of f, it must also satisfy the following inequality

\displaystyle \int_M f dv <0.

This is just a simple observation from integration by parts if we multiply both sides of the PDE by e^{-2u}. Interestingly, Kazdan and Warner were able to show that the above two properties are also sufficient in the sense that it is enough to prove that the PDE is solvable.

In higher dimensions, perhaps, the most natural generalization of the operator \Delta_g is the well-known Paneitz operator and its corresponding Q-curvature, see this link.

Assume that (M,g) is a compact Riemannian 4-manifold without boundary. We denote by P_g^4 the so-called Paneitz operator acting on any smooth function u via the following rule

\displaystyle P_g^4(u) = \Delta _g^2u + {\rm div}\left( {\frac{2}{3}{R_g} - 2{\rm Ric}_g} \right)du ,

where by {\rm Ric} and R we mean the Ricci tensor and the scalar curvature of g, respectively.

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