Surface Motions and Fluid Dynamics *

A certain class of surface motions, including those of a relativistic membrane minimizing the 3-dimensional volume swept out in Minkowski-space, is shown to be equivalent to 3-dimensional steady-state irrotational inviscid isentropic gas-dynamics. The SU(∞) Nahm equations turn out to correspond to motions where the time t at which the surface moves through the point ⇀ r is a harmonic function of the three space-coordinates. This solution also implies the linearisation of a non-trivial-looking scalar field theory. ∗ Talk given at the workshop on ‘Membranes and Higher Dimensional Extended Objects’ ( March 1994, Isaac Newton Institute for Mathematical Sciences) ∗∗ Heisenberg Fellow On leave of absence from the Institute for Theoretical Physics, Karlsruhe University. The dynamics of surfaces is of vital interest to quite a variety of physical problems (see e.g. [1]). Models that have attracted particular attention include the evolution by mean curvature ([2], [3], [4], [5]), the kinetic roughening of growing surfaces and the motion of domain walls [14], developable surfaces and surfaces maintaining constant negative curvature (see e.g. [6], [7]), and in [8], several families of integrable surfaces have been determined, for which the spectral parameter appearing in the zero-curvature condition may be interpreted as a (time-) deformation parameter. Let me consider a time-dependent 2-dimensional surface Σt in R 3, whose motion is locally of the form ⇀̇ r = a · n , (1) where ⇀ r = ⇀ r (t, φ1, φ2) is the position vector, ⇀ n the surface normal, and a = a(g) is assumed to be only a function of the surface area element √ g =| ∂1 ⇀ r × ∂2 ⇀ r | ( ∂1, ∂2 and · denoting differentiation with respect to φ1, φ2 and t, respectively). Two cases of special interest are a = √ g and a = √ 1− g, the former corresponding to a reduction of the self-dual Yang-Mills equations in R4 with the gauge group SDiff Σt (see e.g. [9], [10]), while the latter corresponds to the surface sweeping out a minimal 3-dimensional hypersurface in Minkowski-space (this was derived in [11]). In order to show the equivalence of (1) with 3-dimensional steady-state isentropic irrotational gasdynamics ( and explicitly solve the case a = √ g) let me write (1) in a slightly different way, namely ẋ = γ{y, z} , ẏ = γ{z, x} , ż = γ{x, y} , (2) where γ = a(g) √g and { .,. } denotes the Poisson-bracket ∂ ∂φ ∂ ∂φ − ∂ ∂φ ∂ ∂φ . Choosing the unknown functions x, y and z as the independent(!) variables, t, φ, φ → x = x(t, φ, φ) , x = y(t, φ, φ) , x = z(t, φ, φ) (3) (possible as long as ⇀̇ r 6= 0), and B1 = γ{y, z}, B2 = γ{z, x}, B3 = γ{x, y} , (4) expressed in the new coordinates, as the new dependent variables, such that ∂t ∧ = ⇀ B · ⇀ ∇ , γ · { ·, ⇀ r } ∧ = ⇀ B× ⇀ ∇ , (5)