The Jacobi Spectrum and Global Second Variation of Pansu Spheres

For $n\ge 2$, we establish second differentiability of horizontal perimeter for compact $C^2$ boundaries in ${\mathbb{H}}^n$ 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.