A Liouville-Type Result for a Fourth Order Equation in Conformal Geometry

Abstract

We show that there are no conformal metrics $g=e^{2u}g_{\mathbb {R}^{4}}$ on $\mathbb {R}^{4}$ induced by a smooth function u ≤ C with Δu(x) → 0 as $|x|\to \infty $ having finite volume and finite total Q-curvature, when Q(x) = 1 + A(x) with a negatively definite symmetric 4-linear form A(x) = A(x,x,x,x). Thus, in particular, for suitable smooth, non-constant $f_{0}\le {\max \limits } f_{0}=0$ on a four-dimensional torus any “bubbles” arising in the limit λ↓ 0 from solutions to the problem of prescribed Q-curvature Q = f0 + λ blowing up at a point p0 with dkf0(p0) = 0 for $k=0,\dots ,3$ and with d4f0(p0) < 0 are spherical, similar to the two-dimensional case.