Cauchy's surface area formula in the Heisenberg groups (Q6044138)

Summary: 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 \textit{J.-F. Hwang} et al. [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 4, No. 1, 129--177 (2005; Zbl 1158.53306)] in \(H_1\) and \textit{J.-H. Cheng} et al. [Math. Ann. 337, No. 2, 253--293 (2007; Zbl 1109.35009)] 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.