Use of quadratic differentials for description of defects and textures in liquid crystals and 2 + 1 gravity

Abstract

The theory of measured foliations which is discussed in Part I in connection with the train tracks and meanders is shown to be related to the theory of Jenkins-Strebel quadratic differentials by Hubbard and Masur (Acta Math. 142 (1979) 221). In this work it is demonstrated that this formalism not only provides the adequate description of defects and textures in liquid crystals but also is idealy suited for study of 2 + 1 classical gravity which was initiated in the seminal paper by Deser et al. (Ann. Phys. 152 (1984) 220). Not only their results are reproduced but, in addition, many new results are obtained. In particular, using the results of Rivin (Ann. Math. 139 (1994) 553) the restriction on the total mass of the 2 + 1 Universe is removed. It is shown, that the masses can have only discrete values and, moreover, the theoretically obtained sum rules forbid the existence of some of these values. The dynamics of 2 + 1 gravity which is associated with the dynamics of train tracks (Part I), is reinterpreted in terms of the emerging hyperbolic 3-manifolds. The paper provides a concise introduction to this topic. The discussion of some connections of the obtained results with related physical problems is also provided. These include (but not limited to): string theory, classical and quantum billiards, dynamics of fracture, statics and dynamics of dislocations and disclinations in solids, etc.