Michael Struwe

ETH Zürich

A Liouville-type result for a fourth order equation in conformal geometry

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 \(Δ u(x)→ 0\) as \(|x|→∞\) 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 \(f_0≤\max f_0=0\) on a four-dimensional torus any “bubbles” arising in the limit \(λ↓ 0\) from solutions to the problem of prescribed \(Q\)-curvature \(Q=f_0+λ\) blowing up at a point \(p_0\) with \(d^kf_0(p_0)=0\) for \(k=0,...,3\) and with \(d^4f_0(p_0)<0\) are spherical, similar to the two-dimensional case.