# 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.