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Concentration phenomena for the fractional Q-curvature equation in dimension 3 and fractional Poisson formulas
Journal of the London Mathematical Society ( IF 1.0 ) Pub Date : 2021-01-21 , DOI: 10.1112/jlms.12437
Azahara DelaTorre 1 , María del Mar González 2 , Ali Hyder 3 , Luca Martinazzi 4
Affiliation  

We study the compactness properties of metrics of prescribed fractional Q -curvature of order 3 in R 3 . We will use an approach inspired from conformal geometry, seeing a metric on a subset of R 3 as the restriction of a metric on 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 Φ in 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:

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 - 3阶曲率 in 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 电阻 3 作为度量的限制 电阻 + 4 与消失的四阶 -曲率。我们将展示具有统一有界分数的一系列此类度量 -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 Φ 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)。
更新日期:2021-01-21
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