Pansu 球面的 Jacobi 谱与全局二阶变分
The Jacobi Spectrum and Global Second Variation of Pansu Spheres
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中文总结 AI 辅助
本文证明了 Pansu 球面的 Jacobi 算子谱性质，其 Morse 指标为 1、零化度 2n+1，核为平移速度，从而球面在保体积变分下稳定。
中文摘要 AI 辅助
对于 $n\ge2$，我们建立了 $\mathbb H^n$ 中紧致 $C^2$ 边界水平周长的二阶可微性，以及其仿射二阶变分沿边界到 $C^1$ 向量场的连续延拓。对于单位 Pansu 球面，Hessian 与闭 Jacobi 形式在两个特征极点处一致，包括移动这些极点的变分。我们计算了其 Jacobi 算子的完整 Friedrichs 谱。该算子具有紧预解式，Morse 指标为一，零化度为 $2n+1$；其核恰好由归一化平移速度组成。因此，球面在保体积环境变分下是稳定的。
英文摘要
For $n\ge2$, 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.
发表机构
- National Taiwan University（国立台湾大学）
- Academia Sinica（中央研究院）
机构由 AI 辅助整理，请以论文原文为准。
