On the absence of point defects in biaxial Landau--de Gennes models

Abstract

We study local minimizers of a sextic-potential Landau--de Gennes energy for nematic liquid crystals in the small-elastic-constant limit. Under the uniform energy and $L^{\infty }$ bounds, these minimizers converge to a locally energy-minimizing harmonic map ${\mathbf{Q}}_0$ into a biaxial vacuum manifold. The main result of this paper is that such a limiting map ${\mathbf{Q}}_0$ has no interior point singularities. The proof relies on a geometric identification of the lifted Frobenius metric on the universal cover ${\mathbb{S}}^3$ with a rescaled Berger metric. For a hypothetical tangent cone at a point singularity, its link is a nonconstant harmonic two-sphere into the Berger sphere. We construct a smooth variation field adapted to the Hopf direction and show that it induces a quantitative instability estimate, in contradiction with the shifted stability inequality inherited from local minimality. This replaces the usual round-sphere test fields by a target-specific construction and rules out interior point defects in the biaxial setting. The result is sharp in view of the well-known existence of point defects in the uniaxial theory.

28 pages, comments are welcome!