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The Jacobi Spectrum and Global Second Variation of Pansu Spheres
Abstract
For n≥2, we establish second differentiability of horizontal perimeter for compact C2 boundaries in ℍn and the continuous extension of their affine second variation to C1 vector fields along the boundary. For the unit Pansu sphere, the Hessian agrees with the closed Jacobi form across both characteristic poles, including variations which move them. We compute the full Friedrichs spectrum of its Jacobi operator. The operator has compact resolvent, Morse index one and nullity 2n+1; its kernel consists exactly of normalized translation speeds. Consequently, the sphere is stable under volume-preserving ambient variations.
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BibTeX
@misc{chen2026jacobi,
title = {{The Jacobi Spectrum and Global Second Variation of Pansu Spheres}},
author = {Wei-Chun Chen and Wei-Ting Kao and Chun-Kai Lien},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03899},
eprint = {2610.03899},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03899},
}
RIS
TY - UNPB
TI - The Jacobi Spectrum and Global Second Variation of Pansu Spheres
AU - Wei-Chun Chen
AU - Wei-Ting Kao
AU - Chun-Kai Lien
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03899
UR - https://arxiv.org/abs/2610.03899
AB - For n≥2, we establish second differentiability of horizontal perimeter for compact C^2 boundaries in ℍ^n and the continuous extension of their affine second variation to C^1 vector fields along the boundary. For the unit Pansu sphere, the Hessian agrees with the closed Jacobi form across both characteristic poles, including variations which move them. We compute the full Friedrichs spectrum of its Jacobi operator. The operator has compact resolvent, Morse index one and nullity 2n+1; its kernel consists exactly of normalized translation speeds. Consequently, the sphere is stable under volume-preserving ambient variations.
ER -