A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds (Summary).

Paneitz, Stephen M.

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Let $(M, g)$ be a compact Riemannian manifold of dimension $n \geq 3$ .We suppose that $g$ is a metric in the Sobolev space $H_{2}^{p} (M, T^{*} M \otimes T^{*} M)$ with $p > \frac{n}{2}$ and there exist a point $P \in M$ and $δ > 0$ such that $g$ is smooth in the ball $B_{p} (δ)$ . We define the second Yamabe invariant with singularities as the infimum of the second eigenvalue of the singular Yamabe operator over a generalized class of conformal metrics to $g$ and of volume $1$ . We show that this operator is attained by a generalized metric, we deduce nodal solutions to a Yamabe type equation...