We prove the existence of metrics with prescribed $Q$-curvature under natural assumptions on the sign of the prescribing function and the background metric. In the dimension four case, we also obtain existence results for curvature forms requiring only restrictions on the Euler characteristic. Moreover, we derive a prescription result for open submanifolds which allow us to conclude that any smooth function on $\mathbb{R}^n$ can be realized as the $Q$-curvature of a Riemannian metric.

中文翻译：

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关于黎曼流形中规定的 Q 曲率问题

我们证明了在对指定函数的符号和背景度量的自然假设下具有规定的 $Q$-curvature 的度量的存在。在四维情况下，我们也得到了只需要欧拉特性限制的曲率形式的存在结果。此外，我们推导出开子流形的处方结果，这使我们能够得出结论，$\mathbb{R}^n$ 上的任何平滑函数都可以实现为黎曼度量的 $Q$-curvature。