Singular gradient flow of the distance function and homotopy equivalence

Details Singular gradient flow of the distance function and homotopy equivalence Singular gradient flow of the distance function and homotopy equivalence

2011-09-25

arxiv.org/abs/1109.5375v1

Mathematics

Analysis of PDEs

Scientific paper

It is a generally shared opinion that significant information about the topology of a bounded domain $\Omega $ of a riemannian manifold $M$ is encoded into the properties of the distance, $d_{\partial\Omega}$, %, $d:\Omega\rightarrow [0,\infty [$, from the boundary of $\Omega$. To confirm such an idea we propose an approach based on the invariance of the singular set of the distance function with respect to the generalized gradient flow of of $d_{\partial\Omega}$. As an application, we deduce that such a singular set has the same homotopy type as $\Omega$.

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