Generalized gradient flow and singularities of the Riemannian distance function

Piermarco Cannarsa[1]

  • [1] Dipartimento di Matematica Università di Roma ‘Tor Vergata’ Via della Ricerca Scientifica 1 00133 Roma Italy

Séminaire Laurent Schwartz — EDP et applications (2012-2013)

  • Volume: 2012-2013, page 1-16
  • ISSN: 2266-0607

Abstract

top
Significant information about the topology of a bounded domain Ω of a Riemannian manifold M is encoded into the properties of the distance, d ∂ Ω , from the boundary of Ω . We discuss recent results showing the invariance of the singular set of the distance function with respect to the generalized gradient flow of d ∂ Ω , as well as applications to homotopy equivalence.

How to cite

top

Cannarsa, Piermarco. "Generalized gradient flow and singularities of the Riemannian distance function." Séminaire Laurent Schwartz — EDP et applications 2012-2013 (2012-2013): 1-16. <http://eudml.org/doc/275792>.

@article{Cannarsa2012-2013,
abstract = {Significant information about the topology of a bounded domain $\Omega $ of a Riemannian manifold $M$ is encoded into the properties of the distance, $d_\{\partial \Omega \}$, from the boundary of $\Omega $. We discuss recent results showing the invariance of the singular set of the distance function with respect to the generalized gradient flow of $d_\{\partial \Omega \}$, as well as applications to homotopy equivalence.},
affiliation = {Dipartimento di Matematica Università di Roma ‘Tor Vergata’ Via della Ricerca Scientifica 1 00133 Roma Italy},
author = {Cannarsa, Piermarco},
journal = {Séminaire Laurent Schwartz — EDP et applications},
keywords = {generalized characteristics; semiconcavity; generalized gradient flow},
language = {eng},
pages = {1-16},
publisher = {Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Generalized gradient flow and singularities of the Riemannian distance function},
url = {http://eudml.org/doc/275792},
volume = {2012-2013},
year = {2012-2013},
}

TY - JOUR
AU - Cannarsa, Piermarco
TI - Generalized gradient flow and singularities of the Riemannian distance function
JO - Séminaire Laurent Schwartz — EDP et applications
PY - 2012-2013
PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 2012-2013
SP - 1
EP - 16
AB - Significant information about the topology of a bounded domain $\Omega $ of a Riemannian manifold $M$ is encoded into the properties of the distance, $d_{\partial \Omega }$, from the boundary of $\Omega $. We discuss recent results showing the invariance of the singular set of the distance function with respect to the generalized gradient flow of $d_{\partial \Omega }$, as well as applications to homotopy equivalence.
LA - eng
KW - generalized characteristics; semiconcavity; generalized gradient flow
UR - http://eudml.org/doc/275792
ER -

References

top
  1. P. Albano, P. Cannarsa, Structural properties of singularities of semiconcave functions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 28 (1999), 719–740. Zbl0957.26002MR1760538
  2. P. Albano, P. Cannarsa, Propagation of singularities for solutions of nonlinear first order partial differential equations, Arch. Ration. Mech. Anal. 162 (2002), 1–23. Zbl1043.35052MR1892229
  3. P. Albano, P. Cannarsa, Khai T. Nguyen, C. Sinestrari, Singular gradient flow of the distance function and homotopy equivalence, Math. Ann. 356 (2013), 23–43. Zbl1270.35012MR3038120
  4. G. Alberti, L. Ambrosio, P. Cannarsa, On the singularities of convex functions, Manuscripta Math. 76 (1992), 421–435. Zbl0784.49011MR1185029
  5. D. Attali, J.-D. Boissonnat, H. Edelsbrunner, Stability and computation of medial axes—a state-of-the-art report, in Mathematical Foundations of Scientific Visualization, Computer Graphics, and Massive Data Exploration (G. Farin, H.-C. Hege, D. Hoffman, C.R. Johnson, K. Polthier eds.), 109–125, Springer, Berlin (2009). Zbl1192.68555MR2560510
  6. P. Cannarsa, C. Sinestrari, Semiconcave functions, Hamilton-Jacobi equations, and optimal control. Birkhäuser, Boston (2004). Zbl1095.49003MR2041617
  7. P. Cannarsa,Y. Yu, Singular dynamics for semiconcave functions, J. Eur. Math. Soc. (JEMS) 11 (2009), 999–1024. Zbl1194.35015MR2538498
  8. M.G. Crandall, L.C. Evans, P.L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 282 (1984), 487–502. Zbl0543.35011MR732102
  9. C. Dafermos, Generalized characteristics and the structure of solutions of hyperbolic conservation laws, Indiana Univ. Math. J. 26 (1977), 1097–1119. Zbl0377.35051MR457947
  10. M. Do Carmo, Riemannian geometry, Birkhäuser, Boston (1992). Zbl0752.53001MR1138207
  11. A.Lieutier, Any open bounded subset of ℝ n has the same homotopy type as its medial axis, Comput. Aided Design 36 (2004), 1029–1046. 
  12. P. Petersen, Riemannian geometry, Springer, New York (2006). Zbl1220.53002MR2243772
  13. C. Villani, Optimal transport, old and new. Springer, Berlin - Heidelberg (2009). Zbl1156.53003MR2459454
  14. Y. Yu, A simple proof of the propagation of singularities for solutions of Hamilton-Jacobi equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 5 (2006), 439–444. Zbl1150.35002MR2297718

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.