Integral Menger curvature for surfaces |
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Authors: | Pawe? Strzelecki Heiko von der Mosel |
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Institution: | aInstytut Matematyki, Uniwersytet Warszawski, ul. Banacha 2, PL-02-097 Warsaw, Poland;bInstitut für Mathematik, RWTH Aachen, Templergraben 55, D-52062 Aachen, Germany |
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Abstract: | We develop the concept of integral Menger curvature for a large class of nonsmooth surfaces. We prove uniform Ahlfors regularity and a C1,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 Mp(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) Mp-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 Hölder 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. |
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Keywords: | MSC: 28A75 49Q10 49Q20 53A05 53C21 46E35 |
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