Remarks on the space of volume preserving embeddings

Abstract

Let (N, g) be a Riemannian manifold. For 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 * µ f = µ , where µ f (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 .

Riemannian geometry of the space of volume preserving immersions

“…In the following we will show that Imm * µ (M, N ) is a submanifold of the manifold of immersions. We will follow the proof of [19], but with much less restrictive conditions. Theorem 2.…”

“…Thus, as a first step, we want to ged rid of that additional condition and show a similar statement for the spaces in the above diagram. Similar, as in [19], the proof of this statements will be an application of the Nash-Moser inverse function theorem, however we will have to consider a different splitting of the tangent space. The proof of these statements will be given in Sect.…”

“…In the article [19] it has been shown that the space Emb × µ (M, N ) is a smooth tame splitting submanifold of the space of all smooth embeddings Emb(M, N ), where the elements of the spaces Emb × µ (M, N ) are assumed to have nowhere vanishing second fundamental form. The choice of this space is not very fortunate for our purposes for various reasons; e.g., in the case of closed surfaces in R 3 this condition restricts to convex surfaces only.…”

Uniformly Compressing Mean Curvature Flow

“…We first observe that the geodesics inà k are determined by the condition ∂ τ τ γ ⊥ T γÃk , which can be expressed as (91) ∂ τ τ γ = ς H(γ) + dγ · grad γ * · ς, γ ∈à k , cf. [11,34]. Then we can calculate the Hessian of the L 2 -mass, taking into account (84) (with w = γ and div γ(M) (γ • γ −1 ) = k):…”

“…Our flow is driven by the orthogonal projection of the mean curvature onto T A k . We dub the resulting object the uniformly compressing mean curvature flow (UCMCF) because the evolving surfaces can be thought of as being constituted by fluid particles whose density depends merely on time (the surfaces in question up to a timedependent constant are incompressible membranes [18,20,34]). The UCMCF is by construction the negative gradient flow of the volume functional on A k .…”