2026/07/20 by Jérôme Vétois, Samuel Zeitler
#math.DG #math.AP
Given n,m∈ℕ such that n≥2m≥4, letting g be a conformally Euclidean metric on ℝn, we consider the question of positivity of the lower-order Q-curvatures Qg(2k) for k∈\1,\dotsc,m-1\ when Qg(2m) is assumed to be nonnegative and not identically zero. We assume moreover that the scalar curvature of the metric g is nonnegative near infinity if n=2m or that Qg(2m) satisfies a slow decay barrier condition near infinity if n>2m. Positive results for this question have been obtained by Gursky and Malchiodi for m=2 and k=1 in the context of closed manifolds with nonnegative scalar curvature and by Li and Xu and Li, Wei, and Xu for m≥2 and k∈\1,min(m-1,2)\ in the context of conformally Euclidean metrics on ℝn. These results hold for all n≥2m. Considering the case where m≥4 and k=3, we obtain a positive result for this question when n∈\2m,2m+1,\dotsc,4m-6\, namely for these dimensions, we obtain that if Qg(2m)≥0 and Qg(2m)\not≡0 in ℝn, then Qg(6)>0. On the other hand, in surprising contrast with the results of Gursky and Malchiodi, Li and Xu, and Li, Wei, and Xu, we find that the answer to this question is negative when k=3 and n≥ Nm for some Nm∈ℝ. In this case, we are able to construct examples of conformally Euclidean metrics such that Qg(2m) is positive everywhere, but Qg(6) is negative at some point. By stereographic projection, our examples extend to metrics conformal to the standard metric on \mathbbSn.