Abstract
If U:[0,+infty [times M is a uniformly continuous viscosity solution of the evolution Hamilton-Jacobi equation ∂tU+H(x,∂xU)=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\partial _{t}U+ H(x,\\partial _{x}U)=0, $$\\end{document} 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.
Highlights
Let M be a smooth connected but not necessarily compact manifold
The distance d that we will use on M is the Riemannian distance obtained from the Riemannian metric
Our methods permit to separate the study of these two sets. To state another consequence of Theorem 1.1, we introduce the following definition: Definition 1.2. — If (M, g) is a complete Riemannian manifold, we define the subset U (M, g) ⊂ M × M as the set of (x, y) ∈ M × M such that there exists a unique minimizing g-geodesic between x and y
Summary
Let M be a smooth connected but not necessarily compact manifold. We will assume M endowed with a complete Riemannian metric g. — If (M, g) is a complete Riemannian manifold, we define the subset U (M, g) ⊂ M × M as the set of (x, y) ∈ M × M such that there exists a unique minimizing g-geodesic between x and y. This set U (M, g) contains a neighborhood of the diagonal M ⊂ M × M. If the function U : O → R, defined on the open subset O ⊂ R × M is a continuous viscosity solution of the evolutionary HamiltonJacobi equation (1.1), the set (U) ⊂ O of singularities of U is locally contractible.
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