Abstract
If \(U:[0,+\infty [\times M\) is a uniformly continuous viscosity solution of the evolution Hamilton-Jacobi equation
where \(M\) is a not necessarily compact manifold, and \(H\) is a Tonelli Hamiltonian, we prove the set \(\Sigma (U)\), of points in \(]0,+\infty [\times M\) where \(U\) is not differentiable, is locally contractible. Moreover, we study the homotopy type of \(\Sigma (U)\). 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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Cannarsa, P., Cheng, W. & Fathi, A. Singularities of solutions of time dependent Hamilton-Jacobi equations. Applications to Riemannian geometry. Publ.math.IHES 133, 327–366 (2021). https://doi.org/10.1007/s10240-021-00125-5
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DOI: https://doi.org/10.1007/s10240-021-00125-5