Let $(M,g)$ be a closed Riemannian manifold of dimension $n>7$. In this setting I have been able to prove that the Green's function of a positive Paneitz-Branson operator is non-negative. Furthermore, we can also conclude that the Green's function can't vanish on open sets. I need the Green's function to be positive, though, so I was wondering if anyone had an idea as to how to establish positivity of the Green's function under the hypotheses used above.

The definition of the Paneitz-Branson operator $P_g$ of the metric is as follows:

$P_g:= (\Delta_g)^2 - div_g(a_n R_g g + b_n Ric_g)d + Q_g. $

Here $\Delta_g$ is the Laplace-Beltrami operator, $a_n = \frac{(n-2)^2 +4}{2(n-1)(n-2)}$, $b_n = \frac{-4}{n-2}$, and $Q$ is the $Q$-curvature of the metric $g$. Note that it is a fourth-order operator and hence classical maximum principles can't be used in general. Also, if the hypothesis that $g$ is locally conformally flat is used, that would be ok too. Of course, if a counter-example is found that would be of interest as well.

  • $\begingroup$ +1 for the username $\endgroup$
    – Nik Weaver
    Mar 26 '14 at 6:53

You'll find the answer in a recent and wonderful paper of Matthew J. Gursky and Andrea Malchiodi : http://arxiv.org/abs/1401.3216

  • $\begingroup$ +1 for the username. $\endgroup$ Mar 26 '14 at 12:48

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.