Characterizing $W^{2,p}$~submanifolds by $p$-integrability of global curvatures

Mathematics – Classical Analysis and ODEs

Scientific paper

Rate now

  [ 0.00 ] – not rated yet Voters 0   Comments 0

Details

44 pages, 2 figures

Scientific paper

We give sufficient and necessary geometric conditions, guaranteeing that an immersed compact closed manifold $\Sigma^m\subset \R^n$ of class $C^1$ and of arbitrary dimension and codimension (or, more generally, an Ahlfors-regular compact set $\Sigma$ satisfying a mild general condition relating the size of holes in $\Sigma$ to the flatness of $\Sigma$ measured in terms of beta numbers) is in fact an embedded manifold of class $C^{1,\tau}\cap W^{2,p}$, where $p>m$ and $\tau=1-m/p$. The results are based on a careful analysis of Morrey estimates for integral curvature--like energies, with integrands expressed geometrically, in terms of functions that are designed to measure either (a) the shape of simplices with vertices on $\Sigma$ or (b) the size of spheres tangent to $\Sigma$ at one point and passing through another point of $\Sigma$. Appropriately defined \emph{maximal functions} of such integrands turn out to be of class $L^p(\Sigma)$ for $p>m$ if and only if the local graph representations of $\Sigma$ have second order derivatives in $L^p$ and $\Sigma$ is embedded. There are two ingredients behind this result. One of them is an equivalent definition of Sobolev spaces, widely used nowadays in analysis on metric spaces. The second one is a careful analysis of local Reifenberg flatness (and of the decay of functions measuring that flatness) for sets with finite curvature energies. In addition, for the geometric curvature energy involving tangent spheres we provide a nontrivial lower bound that is attained if and only if the admissible set $\Sigma$ is a round sphere.

No associations

LandOfFree

Say what you really think

Search LandOfFree.com for scientists and scientific papers. Rate them and share your experience with other people.

Rating

Characterizing $W^{2,p}$~submanifolds by $p$-integrability of global curvatures does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Characterizing $W^{2,p}$~submanifolds by $p$-integrability of global curvatures, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Characterizing $W^{2,p}$~submanifolds by $p$-integrability of global curvatures will most certainly appreciate the feedback.

Rate now

     

Profile ID: LFWR-SCP-O-488422

  Search
All data on this website is collected from public sources. Our data reflects the most accurate information available at the time of publication.