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

Silvano Delladio

Bollettino dell'Unione Matematica Italiana (2007)

  • Volume: 10-B, Issue: 1, page 237-252
  • ISSN: 0392-4033

Abstract

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Let γ , τ : [ a , b ] R k + 1 be a couple of Lipschitz maps such that γ = ± | γ | τ almost everywhere in [ a , b ] . Then γ ( [ 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.

How to cite

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Delladio, Silvano. "A Result About $C^2$-Rectifiability of One-Dimensional Rectifiable Sets. Application to a Class of One-Dimensional Integral Currents." Bollettino dell'Unione Matematica Italiana 10-B.1 (2007): 237-252. <http://eudml.org/doc/290405>.

@article{Delladio2007,
abstract = {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.},
author = {Delladio, Silvano},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {237-252},
publisher = {Unione Matematica Italiana},
title = {A Result About $C^2$-Rectifiability of One-Dimensional Rectifiable Sets. Application to a Class of One-Dimensional Integral Currents},
url = {http://eudml.org/doc/290405},
volume = {10-B},
year = {2007},
}

TY - JOUR
AU - Delladio, Silvano
TI - A Result About $C^2$-Rectifiability of One-Dimensional Rectifiable Sets. Application to a Class of One-Dimensional Integral Currents
JO - Bollettino dell'Unione Matematica Italiana
DA - 2007/2//
PB - Unione Matematica Italiana
VL - 10-B
IS - 1
SP - 237
EP - 252
AB - 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.
LA - eng
UR - http://eudml.org/doc/290405
ER -

References

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  1. ANZELLOTTI, G. - SERAPIONI, R., C k -rectifiable sets, J. reine angew. Math., 453 (1994), 1-20. MR1285779
  2. ANZELLOTTI, G. - SERAPIONI, R. - TAMANINI, I., Curvatures, Functionals, Currents, Indiana Univ. Math. J., 39 (1990), 617-669. MR1078733DOI10.1512/iumj.1990.39.39033
  3. DELLADIO, S., Slicing of Generalized Surfaces with Curvatures Measures and Diameter's Estimate, Ann. Polon. Math., LXIV 3 (1996), 267-283. Zbl0865.49029MR1410345DOI10.4064/ap-64-3-267-283
  4. DELLADIO, S., Do Generalized Gauss Graphs Induce Curvature Varifolds?Boll. Un. Mat. Ital., 10-B (1996), 991-1017. Zbl0886.49031MR1430163
  5. DELLADIO, S., The projection of a rectifiable Legendrian set is C 2 -rectifiable: a simplified proof, Proc. Royal Soc. Edinburgh, 133A (2003), 85-96. Zbl1035.53010MR1960048DOI10.1017/S0308210500002298
  6. DELLADIO, S., Taylor's polynomials and non-homogeneous blow-ups, Manuscripta Math., 113, n. 3 (2004), 383-396. Zbl1093.53078MR2129311DOI10.1007/s00229-004-0438-0
  7. DELLADIO, S., Non-homogeneous dilatations of a function graph and Taylor's formula: some results about convergence, Real Anal. Exchange, 29, n. 2 (2003/2004), 1-26. MR2083806DOI10.14321/realanalexch.29.2.0687
  8. FEDERER, H., Geometric Measure Theory, Springer-Verlag1969. Zbl0176.00801MR257325
  9. FU, J.H.G., Some Remarks On Legendrian Rectifiable Currents, Manuscripta Math., 97, n. 2 (1998), 175-187. Zbl0916.53038MR1651402DOI10.1007/s002290050095
  10. FU, J.H.G., Erratum to ``Some Remarks On Legendrian Rectifiable Currents'', Manuscripta Math., 113, n. 3 (2004), 397-401. Zbl1066.53014MR2129312DOI10.1007/s00229-004-0437-1
  11. MATTILA, P., Geometry of sets and measures in Euclidean spaces, Cambridge University Press, 1995. Zbl0819.28004MR1333890DOI10.1017/CBO9780511623813
  12. MORGAN, F., Geometric Measure Theory, a beginner's guide, Academic Press Inc.1988. Zbl0671.49043MR933756
  13. SIMON, L., Lectures on Geometric Measure Theory, Proceedings of the Centre for Mathematical Analysis, Canberra, Australia, 3 (1984). MR756417
  14. STEIN, E.M., Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970. Zbl0207.13501MR290095

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