On hypersurfaces inR n+1 with integral bounds on curvature

Smooth geometric evolutions of hypersurfaces

“…Hence, we can extract a subsequence of smooth hypersurfaces ϕ i = ϕ ti and diffeomorphisms σ i : M → M such that, for a fixed metric h on M , the sequence {ϕ i •σ i } converges in the H 2,p weak topology to an immersion ψ : M → R n+1 . With the arguments of the proof of Theorem 8.1 in [13,27] and keeping into account that in our case we have also the estimates (7.6), it is possible to conclude that actually the convergence is in the C ∞ topology and the limit hypersurface is smooth (see also [22], Prop. 3.4).…”

“…To find limit hypersurfaces, we need the following compactness result of Langer and Delladio [13,27].…”

A result about C3-rectifiability of Lipschitz curves

“…Former achievements in this direction include [Delladio 2001] (a somehow surprising application to differential geometry context), [Anzellotti and Delladio 1995] (an application to Willmore problem) and [Delladio 1997] (an application to a problem introduced in [Bellettini et al 1993]). These last two papers followed the idea by De Giorgi of relaxing the functional with respect to L 1 -convergence of the domains of integration.…”

Generalized unit normal bundles

“…An example of application which we consider significant of how this approach can be fruitful, has been given recently in [14], where generalized Gauss graphs theory has been used to extend a compactness result by Langer [19], concerning sets of hypersurfaces subject to an uniform boundedness condition on the volumes and on the L p norm of the second fundamental forms (if p is sufficiently large). Our interest in this subject is justified by the abundance of applications of unit normal bundles in the general literature and, in particular, along the "boundary" mentioned above.…”