Cauchy’s surface area formula in the Heisenberg groups

Abstract

We show an analogue of Cauchy's surface area formula for the Heisenberg groups ${\mathbb{H}}_n$ for $n\ge 1$, 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 in ${\mathbb{H}}_1$ and Cheng–Hwang– Yang in ${\mathbb{H}}_n$ for $n\ge 2$. 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 choice of the projected directions.