U ( 1 ) -invariant minimal hypersurfaces in R 1 , 3

Two simple ﬁrst order equations are derived, and studied from various points of view, describing the motion of axially symmetric membranes sweeping out time-like zero-mean-curvature manifolds in 4-dimensional Minkowski-space.

explicit examples in R 3,1 (cp. [6,7,11]), apart from the trivial cone [3] and a few algebraic solutions (see [6]) as well as certain melting ice-blocks, respectively scalactites, described in terms of elliptic functions [5]. Here, the main focus will be on two parametric first-order equations governing the motion of axially symmetric membranes, i.e. minimal foliations of (part of) R 3,1 possessing a u(1) symmetry. The existence of transformations (between solutions) similar to those of Bäcklund and Bianchi [8] (for surfaces of constant negative curvature) resp. Thybault (for mininal surfaces [9]) in R 3 is conjectured.
the μ = 0 part of (2) says that the ratio of the determinant of the metric g ab induced from R M+1 on M and G 00 = 1 −˙ x 2 has to be time-independent, while the remaining second order equations, actually follow from (4) and (5), as long as the velocity ˙ x and the tangent vectors ∂ a x, a = 1, . . . , M, are linear independent.
A Hamiltonian formulation of (8)/(9) is given by as can be shown by first noting the reparametrization invariance of H , resp. φ = 0, and then noting that for φ = 0 the Hamiltonian density will actually be time-independent, Expressing (14) in terms oḟ reproduces (9), while φ = 0 gives (8).

Evolution of geometric quantities and singularity formation
Already long ago [5] evolution equations for the first and second fundamental form of were found to bė For axially symmetric surfaces, cp. (7), with are both diagonal, so that (17) reduces tȯ The Ansatz for self-similar solutions of (8)/(9) when t := t 0 − t 0, i.e. near the time t 0 of singularity formation gives 2 > 1 and γ < 1; hence, implying the linear equation resp.

Trigonometric formulation
Noting that (8)/(9) may be written as the signs and convention chosen such that for shrinking convex surfaceṡ Expressing the derivatives of r and z (cp. (29)) as the integrability conditions give

Hydrodynamic formulation and Lax-pair
Applying the hodograph transformation (interchanging independent and dependent variables) t, ϕ → r, z to the equations of motioṅ is the inverse of the determinant of ṙz −żr (= v √ r 2 + z 2 , if using (30)): The integrability condition for (36) (i.e. cross differentiation) yields which is the symmetry reduction t( , of a hydrodynamic formulation found many years ago [12][13][14] for extremal hypersurfaces in terms of the time t at which the hypersurface x(t, ϕ 1 · · · ϕ M=2 ) passes a point x in space It was already indicated in [13] that with γ = 1 ( ∇t) 2 − 1 (40) is equivalent to the commutativity of 2 vector fields, as the identity can be used to prove that in the axially symmetric case, with A corresponding statement actually holds for arbitrary γ .
Although the above can be written in terms of divergence free vector fields, i.e. (via the relation with Poisson brackets) functions of ϕ 1 and ϕ 2 , one does not get any nontrivial conserved quantities, as Tr L n>1 = r n ϕ 1 e inϕ 2 dϕ 1 dϕ 2 ≡ 0. (51) Note that applying the three-dimensional hodograph transformation x 1 x 2 x 3 → ϕ 0 = t, ϕ 1 , ϕ 2 to general (not necessarily normal, or axially symmetric) surface motion, (42) becomes Amazingly it also holds that, for arbitrary γ : For axially symmetric (not necessarily normal) motion (52) reduces to so that can straightforwardly be written in terms of r(t, ϕ) and z(t, ϕ).
Instead of explicitly parametrizing (61) in an analogous way, note that which is the second order equation following from Of interest are also the light-cone-versions, r(t, z) = s(τ = t+z 2 , ζ = t − z), satisfying it is interesting to insert this Ansatz into (72), which yields the non-linear ODE together with a linear ODE determining then B, which integrates tȯ one sees that there is a large class of solution, in terms of elliptic functions, parametrized by δ.
Note that due to (72) being invariant under ζ → ζ + constant, one solution of (75) is B = (const) · A, and the other one (reducing the order of (75), via B = A · E) is given as A times a solution of AË = 4Ė, i.e. (78)

Parametric light-cone description
The perhaps simplest description of axially symmetric mem- This PDE is the compatibility condition foṙ just as is needed for the reconstruction of the hypersurface in R 1,3 .