We study the isoperimetric problem for the axially symmetric sets in the Heisenberg group \(\mathbb {H}^n\) with density \(|z|^p\). At first, we prove the existence of weighted isoperimetric sets. Then, we characterize weighted isoperimetric sets uniquely as bubble sets. Finally, we deduce an interesting result that, up to a constant multiplicator, \(|z|^p\) is the only horizontal radial density for which bubble sets can be weighted isoperimetric sets.

中文翻译：

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海森堡群$$ \ pmb {\ mathbb {H} ^ n} $$ H n的等密度问题

我们研究密度为\（| z | ^ p \）的Heisenberg群\（\ mathbb {H} ^ n \）中的轴对称集的等距问题。首先，我们证明了加权等长集的存在。然后，我们将加权等测集独特地描述为气泡集。最后，我们得出一个有趣的结果，直到一个恒定的乘数，\（| z | ^ p \）是唯一可以将气泡集加权为等渗集的水平径向密度。