Integral Menger curvature for surfaces

By Pawel Strzelecki, H. von der Mosel. 2009

55 citations recorded by the source corpus.

Abstract We develop the concept of integral Menger curvature for a large class of nonsmooth surfaces. We prove uniform Ahlfors regularity and a C 1 , λ -a priori bound for surfaces for which this functional is finite. In fact, it turns out that there is an explicit length scale R > 0 which depends only on an upper bound E for the integral Menger curvature M p ( Σ ) and the integrability exponent p, and not on the surface Σ itself; below that scale, each surface with energy smaller than E looks like a nearly flat disc with the amount of bending controlled by the (local) M p -energy. Moreover, integral Menger curvature can be defined a priori for surfaces with self-intersections or branch points; we prove that a posteriori all such singularities are excluded for surfaces with finite integral Menger curvature. By means of slicing and iterative arguments we bootstrap the Holder exponent λ up to the optimal one, λ = 1 − ( 8 / p ) , thus establishing a new geometric ‘Morrey–Sobolev’ imbedding theorem. As two of the various possible variational applications we prove the existence of surfaces in given isotopy classes minimizing integral Menger curvature with a uniform bound on area, and of area minimizing surfaces subjected to a uniform bound on integral Menger curvature.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a10e70954082ba6e7f03353cedb2395857e13400

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In Applied Mathematics

1. Pawel Strzelecki
2. H. von der Mosel
3. Michael Struwe
4. Changyou Wang
5. Armin Schikorra

In Geometry and Topology

1. Pawel Strzelecki
2. H. von der Mosel

Citations in the corpus

• Tangent-point self-avoidance energies for curves (2010, background): “The reader is invited to fill in the computational details or to consult [25, 27].”
• Regularizing and self-avoidance effects of integral Menger curvature (2010, unknown): “In an ongoing research project we are investigating geometric curvature energies Mp(·) for hypersurfaces; see the upcoming paper [36].”