Exact Hessian Cancellation and Quartic Nondegeneracy at the Round Circle
Abstract
We study the decomposed Möbius energies E1 and E2 at the round circle and determine their complete normal Hessians: H1(C)=2π(|Ds|3−|Ds|)I2 and H2(C)=−(4π/3)(|Ds|3−|Ds|)I2. Thus H1(C)=3HE(C) and H2(C)=−2HE(C), where HE(C) is the Hessian of the full Möbius energy. This exact proportionality singles out the Möbius-invariant combination F=2E1+3E2+2π2. 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 ℝ3, Möbius invariance in ℝ4 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, D4F(C)[u,u,u,u]≥ 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.
Related
- An Exact Counterexample to Affine-Like Price-of-Anarchy Shape in Quartic BPR Routing
- Exact counting of unlabeled quartic graphs by permutation-cycle aggregation
- Exact high-order GPT cancellation in noncircular multilayers
- The Projected Hessian Quantification Theorem: Exact Duality For Constrained Eigenvalues
- Chebyshev-Exact Acceleration under Hessian Variation, II: Computer-Assisted Variational Lower Bound
BibTeX
@misc{ishizeki2026exact,
title = {{Exact Hessian Cancellation and Quartic Nondegeneracy at the Round Circle}},
author = {Aya Ishizeki},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02797},
eprint = {2610.02797},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02797},
}
RIS
TY - UNPB
TI - Exact Hessian Cancellation and Quartic Nondegeneracy at the Round Circle
AU - Aya Ishizeki
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.02797
UR - https://arxiv.org/abs/2610.02797
AB - 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 ℝ^3, Möbius invariance in 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]≥ 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.
ER -