Remarks on the space of volume preserving embeddings

Abstract

Let (N,g) be a Riemannian manifold. Given a compact, connected and oriented submanifold M of N, we define the space of volume preserving embeddings Embμ(M,N) as the set of smooth embeddings f:M↪N such that f⁎μf=μ, where μf (resp. μ) is the Riemannian volume form on f(M) (resp. M) induced by the ambient metric g (the orientation on f(M) being induced by f).In this article, we use the Nash–Moser inverse function Theorem to show that the set of volume preserving embeddings in Embμ(M,N) whose mean curvature is nowhere vanishing forms a tame Fréchet manifold, and determine explicitly the Euler–Lagrange equations of a natural class of Lagrangians.As an application, we generalize the Euler equations of an incompressible fluid to the case of an “incompressible membrane” of arbitrary dimension moving in N.