An Elementary Obstruction to Stability of Hessian Metrics
Manuscript 21 October 2026 · Online 23 October 2026
Abstract
The property of admitting a positive definite Hessian metric is not open under small affine deformations of a compact manifold. On the fixed torus T^2, the classical shear family with sole nonzero Christoffel coefficient Gamma^1_22=t consists of complete, volume-preserving affine connections and converges smoothly to the Euclidean connection. For an arbitrary hypothetical Hessian metric, one component of the Codazzi equation integrates to t times the integral of g_11 being zero, excluding every nonzero t. Products give the same obstruction in all dimensions at least two. We also give an averaging criterion for translation-invariant affine connections on tori. The affine family is due to the classical theory described by Baues and Goldman; this note supplies a self-contained application to Shimizu's stability question, without a novelty or priority claim. This is an unrefereed, self-audited preprint with AI assistance disclosed in the text.
Record
- Affiliation
- Mercury Software GmbH
- Contact
- [email protected] · GitHub
- Result
- Counterexample to openness of Hessian-metric existence
- Categories
- math.DG
- Manuscript
- 21 October 2026
- Online release
- 23 October 2026
- Version
- 1.0
- License
- Creative Commons Attribution 4.0 International
Files and verification
The PDF is the canonical reading copy. The source archive contains the LaTeX manuscript, bibliography, reproducibility material, and audit documents without build artefacts.
Citation
Alper Ferudun, “An Elementary Obstruction to Stability of Hessian Metrics,” EulerSolve Research Papers, AMR-059-0016, 2026. https://doi.org/10.5281/zenodo.23586096.
BibTeX
@misc{ferudun2026amr0590016,
author = {Ferudun, Alper},
title = {An Elementary Obstruction to Stability of Hessian Metrics},
year = {2026},
howpublished = {EulerSolve Research Papers},
url = {https://eulersolve.org/papers/amr-059-0016/},
doi = {10.5281/zenodo.23586096},
note = {AMR-059-0016; unrefereed preprint}
}