Cauchy’s surface area formula in the Heisenberg groups

We show an analogue 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 in \mathbb{H}_1 and Cheng–Hwang– Yang 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 choice of the projected directions.