A Result About $C^2$-Rectifiability of One-Dimensional Rectifiable Sets. Application to a Class of One-Dimensional Integral Currents

Delladio, Silvano

Bollettino della Unione matematica italiana, Série 8, 10B (2007) no. 1, pp. 237-252

Résumé

Let $\gamma, \tau \colon [a, b] \rightarrow R^{k+1}$ be a couple of Lipschitz maps such that $\gamma' = \pm |\gamma'|\tau$ almost everywhere in $[a, b]$. Then $\gamma([a, b])$ is a $C^2$-rectifiable set, namely it may be covered by countably many curves of class $C^2$ embedded in $R^{k+1}$. As a conseguence, projecting the rectifiable carrier of a one-dimensional generalized Gauss graph provides a $C^2$-rectifiable set.