Biharmonic Conformal Immersions of Hypersurfaces: Dirac Rigidity and Mixed Clifford Geometry
Year: 2026
Type: preprint
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 $U=λ^2H$, 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 $\mathbb Rξ\oplus TM$ 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 $$ S^1\times\mathbb S^n\times\mathbb S^n\hookrightarrow\mathbb S^{2n+2} \qquad(n\ge1). $$ Thus the scalar obstruction is sharp with respect to coefficient bundle: the smallest enlargement forced by conformal tension already supports compact non-CMC solutions.
Source: arXiv (Cornell University)
Author: Bhandari, Dipesh
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213581814