Manuscripta Mathematica

, Volume 141, Issue 1, pp 125-147

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

Minimal Hölder regularity implying finiteness of integral Menger curvature

  • Sławomir KolasińskiAffiliated withInstitute of Mathematics, University of Warsaw Email author 
  • , Marta SzumańskaAffiliated withInstitute of Mathematics, University of Warsaw


We study two kinds of integral Menger-type curvatures. We find a threshold value of α 0, a Hölder exponent, such that for all αα 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.

Mathematics Subject Classification (1991)

Primary: 49Q10 Secondary: 28A75 49Q20 49Q15