Cut and singular loci up to codimension 3

Pablo Angulo Ardoy[1]; Luis Guijarro[2]

  • [1] Universidad Autónoma de Madrid Departamento de Matemáticas Facultad de Ciencias Campus de Cantoblanco 28049 Madrid (Spain)
  • [2] Department of Mathematics Universidad Autónoma de Madrid. Please complete ICMAT CSIC-UAM-UCM-UC3M

Annales de l’institut Fourier (2011)

  • Volume: 61, Issue: 4, page 1655-1681
  • ISSN: 0373-0956

Abstract

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We give a new and detailed description of the structure of cut loci, with direct applications to the singular sets of some Hamilton-Jacobi equations. These sets may be non-triangulable, but a local description at all points except for a set of Hausdorff dimension n - 2 is well known. We go further in this direction by giving a classification of all points up to a set of Hausdorff dimension n - 3 .

How to cite

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Ardoy, Pablo Angulo, and Guijarro, Luis. "Cut and singular loci up to codimension 3." Annales de l’institut Fourier 61.4 (2011): 1655-1681. <http://eudml.org/doc/219811>.

@article{Ardoy2011,
abstract = {We give a new and detailed description of the structure of cut loci, with direct applications to the singular sets of some Hamilton-Jacobi equations. These sets may be non-triangulable, but a local description at all points except for a set of Hausdorff dimension $n-2$ is well known. We go further in this direction by giving a classification of all points up to a set of Hausdorff dimension $n-3$.},
affiliation = {Universidad Autónoma de Madrid Departamento de Matemáticas Facultad de Ciencias Campus de Cantoblanco 28049 Madrid (Spain); Department of Mathematics Universidad Autónoma de Madrid. Please complete ICMAT CSIC-UAM-UCM-UC3M},
author = {Ardoy, Pablo Angulo, Guijarro, Luis},
journal = {Annales de l’institut Fourier},
keywords = {Cut locus; Hamilton-Jacobi equations; focal points; cut locus},
language = {eng},
number = {4},
pages = {1655-1681},
publisher = {Association des Annales de l’institut Fourier},
title = {Cut and singular loci up to codimension 3},
url = {http://eudml.org/doc/219811},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Ardoy, Pablo Angulo
AU - Guijarro, Luis
TI - Cut and singular loci up to codimension 3
JO - Annales de l’institut Fourier
PY - 2011
PB - Association des Annales de l’institut Fourier
VL - 61
IS - 4
SP - 1655
EP - 1681
AB - We give a new and detailed description of the structure of cut loci, with direct applications to the singular sets of some Hamilton-Jacobi equations. These sets may be non-triangulable, but a local description at all points except for a set of Hausdorff dimension $n-2$ is well known. We go further in this direction by giving a classification of all points up to a set of Hausdorff dimension $n-3$.
LA - eng
KW - Cut locus; Hamilton-Jacobi equations; focal points; cut locus
UR - http://eudml.org/doc/219811
ER -

References

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  1. G. Alberti, L. Ambrosio, P. Cannarsa, On the singularities of convex functions, Comm. Pure Appl. Math. 76 (1992), 421-435 Zbl0784.49011MR1185029
  2. P. A. Ardoy, L. Guijarro, Balanced split sets and Hamilton Jacobi equations Zbl1221.35105
  3. D. Barden, H. Le, Some consequences of the nature of the distance function on the cut locus in a riemannian manifold, J. London Math. Soc. (2) 56 (1997), 369-383 Zbl0892.53021MR1489143
  4. M. A. Buchner, The structure of the cut locus in dimension less than or equal to six, Compositio Math. 37 (1978), 103-119 Zbl0407.58008MR501100
  5. P. Cannarsa, C. Sinestrari, Semiconcave functions, Hamilton-Jacobi equations, and optimal control, 58 (2004), Birkhäuser Boston, Boston Zbl1095.49003MR2041617
  6. H. Federer, Geometric measure theory, 153 (1969), Springer-Verlag New York Inc., New York Zbl0874.49001MR257325
  7. H. Gluck, D. Singer, Scattering of Geodesic Fields, I, Annals of Mathematics 108 (1978), 347-372 Zbl0399.58011MR506991
  8. J. Hebda, Parallel translation of curvature along geodesics, Trans. Amer. Math. Soc. 299 (1987), 559-572 Zbl0615.53026MR869221
  9. J. Itoh, M. Tanaka, The dimension of a cut locus on a smooth Riemannian manifold, Tohoku Math. J. (2) 50 (1998), 571-575 Zbl0939.53029MR1653438
  10. J. Itoh, M. Tanaka, The Lipschitz continuity of the distance function to the cut locus, Transactions of the A.M.S. 353 (2000), 21-40 Zbl0971.53031MR1695025
  11. YY. Li, L. Nirenberg, The distance function to the boundary, Finsler geometry, and the singular set of viscosity solutions of some Hamilton-Jacobi equations, Comm. Pure Appl. Math. 58 (2005), 85-146 Zbl1062.49021MR2094267
  12. P. L. Lions, Generalized Solutions of Hamilton-Jacobi Equations, 69 (1982), Pitman, Boston, MA Zbl0497.35001MR667669
  13. C. Mantegazza, A. C. Mennucci, Hamilton-Jacobi Equations and Distance Functions on Riemannian Manifolds, Appl. Math. Optim. 47 (2003), 1-25 Zbl1048.49021MR1941909
  14. A.C. Mennucci, Regularity And Variationality Of Solutions To Hamilton-Jacobi Equations. Part I: Regularity (2nd Edition), ESAIM Control Optim. Calc. Var. 13 (2007), 413-417 Zbl1121.49028MR2306644
  15. J. Milnor, Morse theory, 51 (1963), Princeton University Press, Princeton, N.J. Zbl0108.10401MR163331
  16. F. W. Warner, The conjugate locus of a Riemannian manifold, Amer. J. of Math. 87 (1965), 573-604 Zbl0129.36002MR208534

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