论文标题
海森堡组的库奇表面积公式
Cauchy surface area formula in the Heisenberg groups
论文作者
论文摘要
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 $Σ$ in $\mathbb{H}_n$ with its p-normal vector defined almost everywhere on $Σ$ is the average of its projected p-areas onto the orthogonal PANSU球体的所有P正常矢量的补充（达到常数）。该公式提供了$ \ Mathbb {H} _1 $和Cheng-hwang-yang [7]在$ \ Mathbb {h} _n $ for $ n \ geq 2 $中的$ \ mathbb {h} _1 $和Cheng-hwang-yang [7]中定义的P-Ares的几何解释。 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.
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 $Σ$ in $\mathbb{H}_n$ with its p-normal vector defined almost everywhere on $Σ$ 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.
