Some properties of the distance function and a conjecture of De Giorgi

Abstract

In the paper [2] Ennio De Giorgi conjectured that any compact ndimensional regular submanifold M of R n+m , moving by the gradient of the functionalwhere η M is the square of the distance function from the submanifold M and H n is the n-dimensional Hausdorff measure in R n+m , does not develop singularities in finite time provided k is large enough, depending on the dimension n. We prove this conjecture by means of the analysis of the geometric properties of the high derivatives of the distance function from a submanifold of the Euclidean space. In particular, we show some relations with the second fundamental form and its covariant derivatives of independent interest. Proposition 2.2 allows us to write A k in terms of the tensors p k,s and the projections on the tangent and normal spaces (hence contracting with the scalar product of R n+m ), so we get the following corollary.Corollary 2.4. For every k ≥ 3 the symmetric tensor A k can be expressed as a polynomial tensor in B and its covariant derivatives, contracted with the scalar product of R n+m .

Singular perturbations of mean curvature flow

“…for any x ∈ R n+m (we will drop the superscript M when no ambiguity is possible). In this section we recall some facts from [2] and [14] about the distance function and the relations between the high derivatives of η M and the second fundamental form of M.…”

“…We can see then that (1.1) is a singular perturbation of the mean curvature flow, and coincides with it when ε = 0. In [14] (see also [13] and [26]) it is proved that for every ε > 0 the system in (1.1) admits a unique smooth solution defined for all times; we are then interested in the convergence to the mean curvature flow when ε → 0. Our main result is the following.…”

“…c By means of Theorem 4.5 and Theorem 5.9 in [2] and the results of [14], the gradient flow associated with the functional G ε k is given by the PDE system (1.1)…”

Evolution of elastic thin films with curvature regularization via minimizing movements

“…These equations have been already proposed in [19], where (1.9) has been analytically studied. To the best of our knowledge, no analytical results exist in literature for (1.8), unless we restrict ourselves to the case without elasticity, as in [8,9,13,17,39] (see also [5,6]). …”

Ginzburg-Landau relaxation for harmonic maps on planar domains into a general compact vacuum manifold

“…The smoothness of the nearest point retraction is classical [27]. For related computations on the distance function to embedded manifolds, we refer the reader to [3,26]. For every y ∈ N δ N and v ∈ R ν , we have by orthogonality…”