Harmonic maps into round cones and singularities of nematic liquid crystals
This is a study of regularity results for energy minimizing harmonic maps into a round cone. The round cone is the graph \(\mathbb{C}_ \kappa\) of \(| \kappa-1|^{1\over 2}| x|\) with the metric induced from Euclidean space \(\mathbb{R}^ p \times \mathbb{R}\) for \(\kappa \geq 1\) and from Lorentz space \(\mathbb{R}^{p,1}\) for \(0 < \kappa < 1\). Thus \(\mathbb{C}_ \kappa\) is positively curved for \(\kappa > 1\) and negatively curved for \(\kappa < 1\). For a discussion of harmonic maps into singular spaces, in particular, negatively curved spaces, see the important recent preprint, ``Harmonic maps into singular spaces and \(p\)-adic superrigidity for lattices in groups of rank one'' by \textit{M. Gromov} and \textit{R. Schoen}. The present work includes the optimal estimates for the dimension of singularities for minimizing maps into round cones. This completes earlier results in work by the second author [Commun. Pure Appl. Math. 44, No. 4, 453-468 (1991; Zbl 0733.49005)], \textit{L. Ambrosio} [Manuscr. Math. 68, No. 2, 215-228 (1990; Zbl 0713.49005)], and the first author. Minimizers are always Hölder continuous (and even Lipschitz for \(0 < \kappa < 1\)). However, higher regularity is only guaranteed away from the set \(u^{-1}\{0\}\) which has (except for the degenerate case \(u \equiv 0\)) Hausdorff dimension \(\leq m-2\) for \(\kappa > 1\) and dimension \(\leq m-1\) for \(\kappa \leq 1\). Moreover if the cone dimension \(p \geq 3\) and if \(\kappa > 1\), then \(u^{-1}\{0\}\) has dimension \(\leq m - 3\) and is isolated for \(m = 3\). This is shown by studying homogeneous harmonic maps into such cones. For two dimensional domain, these are completely classified with the help of an associated holomorphic quadratic differential. Maps to such cones are applicable for the Ericksen model (involving a variable orientation order) of nematic liquid crystals. By considering the director field as a map into \(\mathbb{R}\mathbb{P}^ 2\) rather than into \(\mathbb{S}^ 2\), the results obtained here are consistent with observed 1 dimensional singularities in nematics.
scientific article; zbMATH DE number 561308
scientific article