Singular gradient flow of the distance function and homotopy equivalence |
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Authors: | P Albano P Cannarsa Khai T Nguyen C Sinestrari |
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Institution: | 1. Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40127, Bologna, Italy 2. Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133, ?Rome, Italy 3. Dipartimento di Matematica Pura ed Applicata, Università di Padova, Via Trieste 63, 35121?, Padua, Italy
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Abstract: | Let $M$ be a Riemannian manifold and let $\varOmega $ be a bounded open subset of $M$ . It is well known that significant information about the geometry of $\varOmega $ is encoded into the properties of the distance, $d_{\partial \varOmega }$ , from the boundary of $\varOmega $ . Here, we show that the generalized gradient flow associated with the distance preserves singularities, that is, if $x_0$ is a singular point of $d_{\partial \varOmega }$ then the generalized characteristic starting at $x_0$ stays singular for all times. As an application, we deduce that the singular set of $d_{\partial \varOmega }$ has the same homotopy type as $\varOmega $ . |
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