Cauchy’s surface area formula in the Heisenberg groups

• Yen-Chang Huang ^[1]
  1. [1] National University of Tainan

     National University of Tainan

     Taiwán

     
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 39, Nº 1, 2023, págs. 165-180
• Idioma: inglés
• DOI: 10.4171/RMI/1320
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• Resumen

  • We show an analogue of Cauchy’s surface area formula for the Heisenberg groups Hn for n 1, which states that the p-area of any compact hypersurface † in Hn with its p-normal vector defined almost everywhere on † 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 H1 and Cheng–Hwang– Yang in Hn for n 2. We also characterize the projected areas for rotationally symmetric domains in Hn; namely, for any rotationally symmetric domain with boundary in Hn, 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.