Diffeomorphism Invariant Integrable Field Theories and Hypersurface Motions in Riemannian Manifolds

TL;DR

This paper explores hypersurface motions in Riemannian manifolds linked to integrable field theories, revealing new solutions and hierarchies with conserved quantities, especially in large-volume membrane limits.

Contribution

It introduces a Hamiltonian framework for hypersurface motions, demonstrating integrability and explicit solutions in large-volume membrane models in R^3.

Findings

Solutions for hypersurface motions in R^3 using electrostatic methods.

Identification of integrable hierarchies with conserved charges.

Linearization of the motion equation as a harmonic function.

Abstract

We discuss hypersurface motions in Riemannian manifolds whose normal velocity is a function of the induced hypersurface volume element and derive a second order partial differential equation for the corresponding time function $\tau(x)$ at which the hypersurface passes the point $x$. Equivalently, these motions may be described in a Hamiltonian formulation as the singlet sector of certain diffeomorphism invariant field theories. At least in some (infinite class of) cases, which could be viewed as a large-volume limit of Euclidean $M$-branesmoving in an arbitrary $M+1$-dimensional Riemannian manifold, the models are integrable: In the time-function formulation the equation becomes linear (with $\tau(x)$ a harmonic function on the embedding Riemannian manifold). We explicitly compute solutions to the large volume limit of Euclidean membrane dynamics in $\Real^3$ by methods used in electrostatics and point out an additional gradient flow structure in $\Real^n$. In the Hamiltonian formulation we discover infinitely many hierarchies of integrable, multidimensional, $N$-component theories possessing infinitely many diffeomorphism invariant, Poisson commuting, conserved charges.