A result about C³-rectifiability of Lipschitz curves

Abstract

1+k be Lipschitz. Our main result provides a sufficient condition, expressed in terms of further accessory Lipschitz maps, for the C 3 -rectifiability of γ 0 ([a, b]).

A class of one-dimensional geometric variational problems with energy involving the curvature and its derivatives: a geometric measure-theoretic approach

“…Hence A covers R j and the conclusion follows from the definition of absolute curvature given in [15] and summarized in Sect. 2.…”

“…by [29,Lemma 11.1]. According to [15], for a one-dimensional C 2 -rectifiable subset R of E, a notion of absolute curvature can be provided as follows. First of all, given a countable family A = {C j } (referred in the sequel as "C 2 -covering of R") of curves of class C 2 embedded in E and such that (2.2) holds, at each density point x of the sets R ∩ C j one can define α A R (x) := absolute curvature of C j at x where j is just any index such that R ∩ C j has density one at x.…”

“…We also mention Proposition 6.3 which, in the case D = 2 and for any given 2-storey Gaussian tower [[G, η, θ]], states a formula for the representation of η in terms of the absolute curvature of the carrier R and its approximate derivative (compare [15] where this notion of curvature has been defined or Sect. 2 below, where it has been recalled).…”

A note on Alberti’s Luzin-type theorem for gradients

“…There, (3.3) amounts to the hypotheses for Whitney extensions of Hölder/Lipschitz functions, which are not automatically verified as in the case of Tietze extension for continuous functions. This is reminiscent of S. Delladio's works (see, e.g., [6]) on the higher-order rectifiability criteria on certain generalised fibre bundles.…”

“…Rectifiable N -currents that are non-C 2 -rectifiable: arbitrary N An important problem in geometric measure theory concerns the C 2 -rectifiability of Legendrian currents, which are natural generalisations of graphs of Gauss maps on hypersurfaces with weaker regularity. Pioneered by Anzellotti-Serapioni [2], studies on the C 2 -rectifiability problem have been carried out by Delladio [5,6] and Fu [11,12], among many other researchers. It was first observed in [12] that the C 2 -rectifiability problem is closely related to Alberti's Theorem 1.1.…”