If \(U:[0,+\infty [\times M\) is a uniformly continuous viscosity solution of the evolution Hamilton-Jacobi equation

$$ \partial _{t}U+ H(x,\partial _{x}U)=0, $$

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.

中文翻译：

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时间相关的 Hamilton-Jacobi 方程解的奇异性。在黎曼几何中的应用

如果\(U:[0,+\infty [\times M\)是演化 Hamilton-Jacobi 方程的一致连续粘度解

$$ \partial _{t}U+ H(x,\partial _{x}U)=0, $$

其中\(M\)不一定是紧流形，\(H\)是托内利哈密顿量，我们证明\(\Sigma (U)\)中点的集合\(\Sigma (U)\) ，位于 \(]0,+\infty [ \times M\)其中\(U\)不可微，是局部可收缩的。此外，我们研究了\(\Sigma (U)\)的同伦类型。我们还将距离函数的奇点应用于完整黎曼流形的闭子集。