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Journal Articles Mathematische Annalen Year : 2013

Singular gradient flow of the distance function and homotopy equivalence

Abstract

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 $\Omega$.

Dates and versions

hal-00724729 , version 1 (22-08-2012)

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Paolo Albano, Piermarco Cannarsa, Khai Tien Nguyen, Carlo Sinestrari. Singular gradient flow of the distance function and homotopy equivalence. Mathematische Annalen, 2013, 356 (1), pp.23-43. ⟨10.1007/s00208-012-0835-8⟩. ⟨hal-00724729⟩
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