Abstract
We study two kinds of integral Menger-type curvatures. We find a threshold value of α 0, a Hölder exponent, such that for all α > α 0 embedded C 1,α manifolds have finite curvature. We also give an example of a \({C^{1,\alpha_0}}\) injective curve and higher dimensional embedded manifolds with unbounded curvature.
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Acknowledgments
The first authorwas partially supported by the Polish Ministry of Science grant no. N N201 611140 (years 2011–2012). The second author was partially supported by the joint German–Polish project “Geometric curvature energies”. The authors are indebted to Prof. P. Strzelecki for his valuable suggestions. The second author would like to thank Jonas Azzam for his questions, ideas and fruitful discussions.
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Kolasiński, S., Szumańska, M. Minimal Hölder regularity implying finiteness of integral Menger curvature. manuscripta math. 141, 125–147 (2013). https://doi.org/10.1007/s00229-012-0565-y
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DOI: https://doi.org/10.1007/s00229-012-0565-y