Singularities of spacelike constant mean curvature surfaces in Lorentz–Minkowski space

We study singularities of spacelike, constant (non-zero) mean curvature (CMC) surfaces in the Lorentz–Minkowski 3-space L3. We show how to solve the singular Björling problem for such surfaces, which is stated as follows: given a real analytic null-curve f0(x), and a real analytic null vector field v(x) parallel to the tangent field of f0, find a conformally parameterized (generalized) CMC H surface in L3 which contains this curve as a singular set and such that the partial derivatives fx and fy are given by df0/dx and v along the curve. Within the class of generalized surfaces considered, the solution is unique and we give a formula for the generalized Weierstrass data for this surface. This gives a framework for studying the singularities of non-maximal CMC surfaces in L3. We use this to find the Björling data – and holomorphic potentials – which characterize cuspidal edge, swallowtail and cuspidal cross cap singularities.

CALCULATION OF THE FLORY EXPONENTS BY A SELF-CONSISTENT METHOD

The polymers that have no ends and interact with a background tangent field u(x, t) are considered to act as the random walk subjected to a thermal noise η(t) and to a divergenceless random force u(x, t). Batchelor's suggestion for the diffusive behaviors of a single particle immersed in turbulent flows in extended to the case of polymers conformation. Based upon the assumption that the particle displacement has the Gaussian distribution for the long diffusion time, Flory's exponents are calculated to be ν = 1/2 for d ≥ 4 and ν = 3/(d+2) for d ≤ 4, in agreement with that of Kamien by the dynamical renormalization group method.