Generalized gradient flow and singularities of the Riemannian distance function
Piermarco Cannarsa1
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), Talk no. 9, 16 p.

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.

DOI: 10.5802/slsedp.37
Piermarco Cannarsa 1

1 Dipartimento di Matematica Università di Roma ‘Tor Vergata’ Via della Ricerca Scientifica 1 00133 Roma Italy 
@article{SLSEDP_2012-2013____A9_0,
     author = {Piermarco Cannarsa},
     title = {Generalized gradient flow and singularities of the {Riemannian} distance function},
     journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
     note = {talk:9},
     pages = {1--16},
     publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2012-2013},
     doi = {10.5802/slsedp.37},
     language = {en},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.37/}
}
TY  - JOUR
AU  - Piermarco Cannarsa
TI  - Generalized gradient flow and singularities of the Riemannian distance function
JO  - Séminaire Laurent Schwartz — EDP et applications
N1  - talk:9
PY  - 2012-2013
SP  - 1
EP  - 16
PB  - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
UR  - https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.37/
DO  - 10.5802/slsedp.37
LA  - en
ID  - SLSEDP_2012-2013____A9_0
ER  - 
%0 Journal Article
%A Piermarco Cannarsa
%T Generalized gradient flow and singularities of the Riemannian distance function
%J Séminaire Laurent Schwartz — EDP et applications
%Z talk:9
%D 2012-2013
%P 1-16
%I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
%U https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.37/
%R 10.5802/slsedp.37
%G en
%F SLSEDP_2012-2013____A9_0
Piermarco Cannarsa. Generalized gradient flow and singularities of the Riemannian distance function. Séminaire Laurent Schwartz — EDP et applications (2012-2013), Talk no. 9, 16 p. doi : 10.5802/slsedp.37. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.37/

[1] P. Albano, P. Cannarsa, Structural properties of singularities of semiconcave functions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 28 (1999), 719–740. | Numdam | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[4] G. Alberti, L. Ambrosio, P. Cannarsa, On the singularities of convex functions, Manuscripta Math. 76 (1992), 421–435. | MR | Zbl

[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). | MR | Zbl

[6] P. Cannarsa, C. Sinestrari, Semiconcave functions, Hamilton-Jacobi equations, and optimal control. Birkhäuser, Boston (2004). | MR | Zbl

[7] P. Cannarsa,Y. Yu, Singular dynamics for semiconcave functions, J. Eur. Math. Soc. (JEMS) 11 (2009), 999–1024. | MR | Zbl

[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. | MR | Zbl

[9] C. Dafermos, Generalized characteristics and the structure of solutions of hyperbolic conservation laws, Indiana Univ. Math. J. 26 (1977), 1097–1119. | MR | Zbl

[10] M. Do Carmo, Riemannian geometry, Birkhäuser, Boston (1992). | MR | Zbl

[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). | MR | Zbl

[13] C. Villani, Optimal transport, old and new. Springer, Berlin - Heidelberg (2009). | MR | Zbl

[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. | Numdam | MR | Zbl

Cited by Sources: