DIFFEOMORPHISM INVARIANT INTEGRABLE FIELD-THEORIES AND HYPERSURFACE MOTIONS IN RIEMANNIAN-MANIFOLDS

We discuss hypersurface motions in Riemannian manifolds whose normal v elocity is a function of the induced hypersurface volume element and d erive a second-order partial differential equation for the correspondi ng time function tau(x) at which the hypersurface passes the point x. Equivalently, these motions may be described in a Hamiltonian formulat ion as the singlet sector of certain diffeomorphism-invariant held the ories. At least in some (infinite class of) cases, which could be view ed as a large-volume limit of Euclidean M-branes moving in an arbitrar y M+1-dimensional Riemannian manifold, the models are integrable: In t he time-function formulation the equation becomes linear [with tau(x) a harmonic function on the embedding Riemannian manifold]. We explicit ly compute solutions to the large volume limit of Euclidean membrane d ynamics in R-3 by methods used in electrostatics and point out an addi tional gradient how structure in R-n. In the Hamiltonian formulation w e discover infinitely many hierarchies of integrable, multidimensional , N-component theories possessing infinitely many diffeomorphism invar iant, Poisson commuting, conserved charges. (C) 1998 American Institut e of Physics. [S0022-2488(97)00912-2].