We show the analogy of Cauchy's surface area formula for the Heisenberg groups $\mathbb{H}_n$ for $n\geq 1$, which states that the p-area of any compact hypersurface $\Sigma$ in $\mathbb{H}_n$ with its p-normal vector defined almost everywhere on $\Sigma$ is the average of its projected p-areas onto the orthogonal complements of all p-normal vectors of the Pansu spheres (up to a constant). The formula provides a geometric interpretation of the p-areas defined by Cheng-Hwang-Malchiodi-Yang [9] in $\mathbb{H}_1$ and Cheng-Hwang-Yang [7] in $\mathbb{H}_n$ for $n\geq 2$. We also characterize the projected areas for rotationally symmetric domains in $\mathbb{H}_n$, namely, for any rotationally symmetric domain with boundary in $\mathbb{H}_n$, its projected p-area onto the orthogonal complement of any normal vector of the Pansu spheres is a constant, independent of the choices of the projected directions.

中文翻译：

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海森堡群中的柯西表面积公式

我们展示了海森堡群 $\mathbb{H}_n$ 对于 $n\geq 1$ 的柯西表面积公式的类比，它表明任何紧致超曲面 $\Sigma$ 在 $\mathbb{H 中的 p 面积}_n$ 及其 p 法线向量几乎在 $\Sigma$ 上的任何位置都定义，是其投影到 Pansu 球体的所有 p 法线向量的正交补上的 p 面积的平均值（直到一个常数）。该公式为 $\mathbb{H}_1$ 中的 Cheng-Hwang-Malchiodi-Yang [9] 和 $\mathbb{H}_n$ 中的 Cheng-Hwang-Yang [7] 定义的 p 面积提供了几何解释对于 $n\geq 2$。我们还描述了 $\mathbb{H}_n$ 中旋转对称域的投影区域，即对于边界在 $\mathbb{H}_n$ 中的任何旋转对称域，其投影 p 区域到任意Pansu 球体的法向量是一个常数，