Biharmonic Conformal Immersions of Hypersurfaces: Dirac Rigidity and Mixed Clifford Geometry
Abstract
A biharmonic conformal immersion asks whether a hypersurface can become a critical point of the bienergy after a conformal change of the domain metric. We study when this flexibility can produce compact nonconstant-mean-curvature geometry in space forms. For surfaces in three-dimensional space forms, with U=λ2H, Ou's system is encoded exactly by a Laplace-type operator acting on a restricted ambient Killing spinor. We classify all monic scalar--chiral factorizations of this operator. In nonzero curvature such a factorization exists locally exactly when the principal curvatures are locally constant; the normalized Euclidean branch is equally rigid, and the remaining real spherical non-CMC branch is excluded by a finite-type Frobenius obstruction. For m>2, this scalar rigidity does not imply geometric nonexistence. The conformal tension has independent normal and tangential components. We prove that ℝξ⊕ TM is the minimal natural O(m)-equivariant carrier, derive the corresponding strongly elliptic Jacobi block and its two-grade Clifford realization, and identify off-diagonal hybridization as the escape mechanism. Exact cohomogeneity-one reductions, combined with validated interval and Chebyshev/radii-polynomial arguments, yield embedded proper non-CMC, nowhere-umbilical examples in round spheres, including S1×𝕊n×𝕊n↪𝕊2n+2 (n≥1). Thus the scalar obstruction is sharp with respect to coefficient bundle: the smallest enlargement forced by conformal tension already supports compact non-CMC solutions.
Related
- Conformal structures and rigidity of complete stable minimal hypersurfaces
- Biharmonic hypersurfaces in space forms
- Biconservative and Biharmonic Hypersurfaces in Product Spaces
- Minimal immersions by first eigenfunctions of Riemann surfaces with large conformal automorphism groups. The Klein quartic
- Allen--Cahn and mean curvature flows from Clifford hypersurfaces to equators
BibTeX
@misc{bhandari2026biharmonic,
title = {{Biharmonic Conformal Immersions of Hypersurfaces: Dirac Rigidity and Mixed Clifford Geometry}},
author = {Dipesh Bhandari},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.20197},
eprint = {2609.20197},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20197},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Biharmonic Conformal Immersions of Hypersurfaces: Dirac Rigidity and Mixed Clifford Geometry
AU - Dipesh Bhandari
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2609.20197
UR - https://arxiv.org/abs/2609.20197
AB - A biharmonic conformal immersion asks whether a hypersurface can become a critical point of the bienergy after a conformal change of the domain metric. We study when this flexibility can produce compact nonconstant-mean-curvature geometry in space forms. For surfaces in three-dimensional space forms, with U=λ^2H, Ou's system is encoded exactly by a Laplace-type operator acting on a restricted ambient Killing spinor. We classify all monic scalar--chiral factorizations of this operator. In nonzero curvature such a factorization exists locally exactly when the principal curvatures are locally constant; the normalized Euclidean branch is equally rigid, and the remaining real spherical non-CMC branch is excluded by a finite-type Frobenius obstruction. For m>2, this scalar rigidity does not imply geometric nonexistence. The conformal tension has independent normal and tangential components. We prove that ℝξ⊕ TM is the minimal natural O(m)-equivariant carrier, derive the corresponding strongly elliptic Jacobi block and its two-grade Clifford realization, and identify off-diagonal hybridization as the escape mechanism. Exact cohomogeneity-one reductions, combined with validated interval and Chebyshev/radii-polynomial arguments, yield embedded proper non-CMC, nowhere-umbilical examples in round spheres, including S^1×𝕊^n×𝕊^n↪𝕊^2n+2 (n≥1). Thus the scalar obstruction is sharp with respect to coefficient bundle: the smallest enlargement forced by conformal tension already supports compact non-CMC solutions.
ER -