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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Albano, P., Cannarsa, P., Nguyen, K.T. et al. Singular gradient flow of the distance function and homotopy equivalence. Math. Ann. 356, 23–43 (2013). https://doi.org/10.1007/s00208-012-0835-8
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DOI: https://doi.org/10.1007/s00208-012-0835-8