Submitted Paper
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Inserted: 13 nov 2026
Last Updated: 13 nov 2026

Year: 2026

Abstract:

We study complete critical metrics for the quadratic scalar curvature functional $ g\mapsto\int_M S_g^2 dv_g, $ with criticality understood under compactly supported variations, focusing on rigidity phenomena and nontrivial examples. In dimension three, we prove that every complete critical metric is scalar-flat, without any additional assumption. In dimension four, we construct complete non-Einstein critical metrics with bounded, nonconstant negative scalar curvature. In every dimension $n\geq5$, we show that either $S_g\equiv 0$ or $S_g<0$ everywhere, and a new argument proves scalar-flatness whenever $S_g \in L^p$ for some $0<p\leq P_n$, where $P_n>2$ is an explicit threshold. Moreover, we construct complete critical metrics with bounded, negative scalar curvature, decaying to zero at infinity. These examples establish the optimality of our rigidity threshold $P_n$ in dimension six and in every dimension $n\geq10$. To the best of our knowledge, these are the first complete critical metrics for this functional that are neither scalar-flat nor Einstein.

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