# 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

## Access Full Article

top## Abstract

top## How to cite

topCannarsa, 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- P. Albano, P. Cannarsa, Structural properties of singularities of semiconcave functions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 28 (1999), 719–740. Zbl0957.26002MR1760538
- 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
- 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
- G. Alberti, L. Ambrosio, P. Cannarsa, On the singularities of convex functions, Manuscripta Math. 76 (1992), 421–435. Zbl0784.49011MR1185029
- 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
- P. Cannarsa, C. Sinestrari, Semiconcave functions, Hamilton-Jacobi equations, and optimal control. Birkhäuser, Boston (2004). Zbl1095.49003MR2041617
- P. Cannarsa,Y. Yu, Singular dynamics for semiconcave functions, J. Eur. Math. Soc. (JEMS) 11 (2009), 999–1024. Zbl1194.35015MR2538498
- 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
- C. Dafermos, Generalized characteristics and the structure of solutions of hyperbolic conservation laws, Indiana Univ. Math. J. 26 (1977), 1097–1119. Zbl0377.35051MR457947
- M. Do Carmo, Riemannian geometry, Birkhäuser, Boston (1992). Zbl0752.53001MR1138207
- A.Lieutier, Any open bounded subset of ${\mathbb{R}}^{n}$ has the same homotopy type as its medial axis, Comput. Aided Design 36 (2004), 1029–1046.
- P. Petersen, Riemannian geometry, Springer, New York (2006). Zbl1220.53002MR2243772
- C. Villani, Optimal transport, old and new. Springer, Berlin - Heidelberg (2009). Zbl1156.53003MR2459454
- 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 ?

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