Cauchy surface area formula in the Heisenberg groups

TL;DR

This paper extends Cauchy's surface area formula to Heisenberg groups, providing a geometric interpretation of p-areas and characterizing projected areas for rotationally symmetric domains.

Contribution

It introduces an analogue of Cauchy's surface area formula in Heisenberg groups, linking p-areas to projections onto Pansu spheres and characterizing symmetric domain areas.

Findings

p-area equals average of projected p-areas onto Pansu sphere normals

Projected p-area is constant for rotationally symmetric domains

Provides geometric interpretation of p-areas in Heisenberg groups

Abstract

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.