AI 中文总结
该研究证明小弹性常数极限下双轴六次势Landau--de Gennes能量的局部极小元对应的极限调和映射无内点奇点，通过几何等同与变分场构造导出矛盾，弥补了单轴理论存在点缺陷的差异。
AI 中文摘要
我们研究小弹性常数极限下向列相液晶的六次势Landau--de Gennes能量的局部极小元。在一致能量界和$L^\infty$界下，这些极小元收敛到取值于双轴真空流形的局部能量极小调和映射$\mathbf{Q}_0$。本文的主要结果是，此类极限映射$\mathbf{Q}_0$不存在内点奇点。证明依赖于将万有覆叠$\mathbb{S}^3$上的提升Frobenius度量与缩放后的Berger度量进行几何等同。对于点奇点处假设的切锥，其链环是映入Berger球面的非常值调和2-球面。我们构造了适配于Hopf方向的光滑变分场，并证明它诱导出定量的不稳定性估计，这与由局部极小性继承的平移稳定性不等式相矛盾。该方法用针对靶空间的构造替代了通常的球面测试场，排除了双轴情形下的内点缺陷。鉴于单轴理论中已知存在点缺陷，这一结果是紧的。
英文摘要
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.
Comments28 pages, comments are welcome!
