发表机构
Saitama University（埼玉大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究圆环处分解的 Möbius 能量，证明其组合 F 的 Hessian 精确成比例，且第四变分在法向变分上非负，揭示由四阶导数决定的非退化性。
AI 中文摘要
我们研究了分解的 Möbius 能量 $E_1$ 和 $E_2$ 在圆环处的性质，并确定了它们的完整正规 Hessian：$H_1(C)=2\pi(|D_s|^3-|D_s|)I_2$ 和 $H_2(C)=-(4\pi/3)(|D_s|^3-|D_s|)I_2$。因此 $H_1(C)=3H_E(C)$ 且 $H_2(C)=-2H_E(C)$，其中 $H_E(C)$ 是完整 Möbius 能量的 Hessian。这种精确的比例关系凸显了 Möbius 不变的组合 $F=2E_1+3E_2+2\pi^2$。我们证明了 $F(C)=0$ 且其前三阶导数在参数化变分的全空间上消失。对于标量副法线变分，第四变分具有一个傅里叶张量，其具有完整的 $3+1$ 共振消去和有限的 $2+2$ 固定和块。它们的正性由显式的加权差分三对角化和闭式行列式公式得出。对于 $\mathbb{R}^3$ 中的任意法向变分，$\mathbb{R}^4$ 中的 Möbius 不变性将径向分量约化为第二副法线方向。由此产生的双分量四次形式分解为迹、对称无迹和反对称块，它们在活跃坐标上是正的。因此，对于每个光滑实法向场 $u$，$D^4F(C)[u,u,u,u]\geq 0$，且等号恰好发生在圆环的 Möbius 族的法向切空间上。这产生了一个四阶非退化现象，该现象并非仅由总能量 Hessian 决定。
英文摘要
We study the decomposed Möbius energies $E_1$ and $E_2$ at the round circle and determine their complete normal Hessians: $H_1(C)=2π(|D_s|^3-|D_s|)I_2$ and $H_2(C)=-(4π/3)(|D_s|^3-|D_s|)I_2$. Thus $H_1(C)=3H_E(C)$ and $H_2(C)=-2H_E(C)$, where $H_E(C)$ is the Hessian of the full Möbius energy. This exact proportionality singles out the Möbius-invariant combination $F=2E_1+3E_2+2π^2$. We prove that $F(C)=0$ and that its first three derivatives vanish on the full space of parametrized variations. For scalar binormal variations, the fourth variation has a Fourier tensor with complete $3+1$ resonance cancellation and finite $2+2$ fixed-sum blocks. Their positivity follows from an explicit weighted-difference tridiagonalization and a closed determinant formula. For arbitrary normal variations in $\mathbb{R}^3$, Möbius invariance in $\mathbb{R}^4$ reduces the radial component to a second binormal direction. The resulting two-component quartic form splits into trace, symmetric-traceless, and antisymmetric blocks, which are positive on the active coordinates. Consequently, $D^4F(C)[u,u,u,u]\geq 0$ for every smooth real normal field $u$, with equality exactly on the normal tangent space of the Möbius family of round circles. This yields a fourth-order nondegeneracy phenomenon that is not determined by the total-energy Hessian alone.
Comments30 pages. Revised presentation, added details to the determinant argument, and clarified conventions; mathematical results unchanged
