A result about \(C^2\)-rectifiability of one-dimensional rectifiable sets. Application to a class of one-dimensional integral currents. (English) Zbl 1178.53003
Summary: Let \(\gamma,\tau : [a,b] \to \mathbb{R}^{k+1}\) 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 \(\mathbb{R}^{k+1}\). As a consequence, projecting the rectifiable carrier of a one-dimensional generalized Gauss graph provides a \(C^2\)-rectifiable set.
MSC:
53A07 | Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces |
49Q15 | Geometric measure and integration theory, integral and normal currents in optimization |
53C40 | Global submanifolds |