Singularities of solutions of time dependent Hamilton-Jacobi equations. Applications to Riemannian geometry

Piermarco Cannarsa Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133, Roma, Italy Wei Cheng Department of Mathematics, Nanjing University, Nanjing, 210093, China Albert Fathi Georgia Institute of technology & ENS de Lyon (Emeritus), School of Mathematics, Atlanta, GA, 30332, USA

Differential Geometry mathscidoc:2203.10002

Publications mathématiques de l'IHÉS, 133, 327-366, 2021.7
If 𝑈:[0,+∞[×𝑀 is a uniformly continuous viscosity solution of the evolution Hamilton-Jacobi equation ∂_𝑡𝑈+𝐻(𝑥,∂_𝑥 𝑈)=0, where 𝑀 is a not necessarily compact manifold, and 𝐻 is a Tonelli Hamiltonian, we prove the set Σ(𝑈), of points in ]0,+∞[×𝑀 where 𝑈 is not differentiable, is locally contractible. Moreover, we study the homotopy type of Σ(𝑈). We also give an application to the singularities of the distance function to a closed subset of a complete Riemannian manifold.
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@inproceedings{piermarco2021singularities,
  title={Singularities of solutions of time dependent Hamilton-Jacobi equations. Applications to Riemannian geometry},
  author={Piermarco Cannarsa, Wei Cheng, and Albert Fathi},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220316103456927870972},
  booktitle={Publications mathématiques de l'IHÉS},
  volume={133},
  pages={327-366},
  year={2021},
}
Piermarco Cannarsa, Wei Cheng, and Albert Fathi. Singularities of solutions of time dependent Hamilton-Jacobi equations. Applications to Riemannian geometry. 2021. Vol. 133. In Publications mathématiques de l'IHÉS. pp.327-366. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220316103456927870972.
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