Harmonic maps into round cones and singularities of nematic liquid crystals
type: Item
instance of: scholarly article
title: Harmonic maps into round cones and singularities of nematic liquid crystals
zbMATH Open document ID: 0790.58010
DOI: 10.1007/BF03025739
published in: Mathematische Zeitschrift
publication date: 1994-06-30T00:00:00Z
full work available at URL: https://eudml.org/doc/174547
review text: 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 \(\mathbb{S}^ 2\),…
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 49J10
Mathematics Subject Classification ID: 49K10
zbMATH DE Number: 561308
zbMATH Keywords: regularity
zbMATH Keywords: energy minimizing harmonic maps
zbMATH Keywords: round cone
zbMATH Keywords: nematic liquid crystals
zbMATH Keywords: singularities
author: Robert Hardt
author: Fanghua Lin
reviewed by: Robert Hardt
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 561308