Mean curvature flow of a hyperbolic surface

• Published in: Journal of experimental and theoretical physics 2011-12, Vol.113 (6), p.1063-1070
• Main Authors: Ovchinnikov, Yu. N., Sigal, I. M.
• Format: Article
• Language: English
• Subjects: Classical and Quantum Gravitation; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CYLINDRICAL CONFIGURATION; DEFORMATION; DIFFERENTIAL EQUATIONS; DISTANCE; DISTURBANCES; Elementary Particles; MATHEMATICAL SOLUTIONS; Nonlinear; Particle and Nuclear Physics; Physics; Physics and Astronomy; Quantum Field Theory; Relativity Theory; Soft Matter Physics; Solid State Physics; Statistical; SURFACES; TOPOLOGY
• ISSN: 1063-7761; 1090-6509
• DOI: 10.1134/S1063776111140068

A four-parameter family of self-similar solutions is obtained to the mean curvature flow equation for a surface. This family is shown to be stable with respect to a small deformation of a hyperbolic surface. At time instant t *, a singular point is formed within a finite time interval, that is accompanied by a change in the topology of the surface. The solution is continued beyond the singular point. A relationship between the parameters describing the hyperbolic surface before and after the change in the surface topology is obtained. A particular case is analyzed when the unperturbed surface is a cylinder. A cylindrical surface is weakly unstable with respect to a perturbation in the form of a “wide neck.” At the final stage of the development of the neck when its transverse size becomes much less than the cylinder radius at large distances from the neck, the surface flow in a wide region in the neighborhood of the neck is described by a universal two-parameter family of self-similar solutions. These solutions are stable with respect to small perturbations of the surface.