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 $U={\lambda }^{2}H$, 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 $\mathbb{R}\xi \oplus TM$ 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 $$S^1\times {\mathbb{S}}^n\times {\mathbb{S}}^n\hookrightarrow {\mathbb{S}}^{2n+2}(n\ge 1).$$ Thus the scalar obstruction is sharp with respect to coefficient bundle: the smallest enlargement forced by conformal tension already supports compact non-CMC solutions.

arXiv comment67 pages