We study the compactness properties of metrics of prescribed fractional $Q$-curvature of order 3 in ${\mathbb{R}}^3$. We will use an approach inspired from conformal geometry, seeing a metric on a subset of ${\mathbb{R}}^3$ as the restriction of a metric on ${\mathbb{R}}_{+}^4$ with vanishing fourth-order $Q$-curvature. We will show that a sequence of such metrics with uniformly bounded fractional $Q$-curvature can blow up on a large set (roughly, the zero set of the trace of a non-positive bi-harmonic function $\mathrm{\Phi }$ in ${\mathbb{R}}_{+}^4$), in analogy with a four-dimensional result of Adimurthi–Robert–Struwe, and construct examples of such behaviour. In doing so, we produce general Poisson-type representation formulas (also for higher dimension), which are of independent interest.

中文翻译：

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3维分数阶Q曲率方程和分数阶泊松公式的集中现象

我们研究了规定分数的度量的紧凑性 $问$- 3阶曲率 in ${\mathbb{电阻}}^3$. 我们将使用一种受共形几何启发的方法，在一个子集上查看度量${\mathbb{电阻}}^3$ 作为度量的限制 ${\mathbb{电阻}}_{+}^4$ 与消失的四阶 $问$-曲率。我们将展示具有统一有界分数的一系列此类度量$问$-curvature 可以在大集合上爆炸（粗略地说，非正双调和函数的迹的零集 $\mathrm{\Phi }$ 在 ${\mathbb{电阻}}_{+}^4$)，与 Adimurthi-Robert-Struwe 的四维结果类比，并构建此类行为的示例。在这样做时，我们产生了具有独立意义的一般泊松型表示公式（也适用于更高维度）。