Let M be a Riemannian manifold and let Ω be a bounded open subset of M. It is well known that significant information about the geometry of Ω is encoded into the properties of the distance, d∂Ω , from the boundary of Ω. Here, we show that the generalized gradient flow associated with the distance preserves singularities, that is, if x0 is a singular point of d∂Ω then the generalized characteristic starting at x0 stays singular for all times. As an application, we deduce that the singular set of d∂Ω has the same homotopy type as Ω.
Singular gradient flow of the distance function and homotopy equivalence / Paolo Albano; Piermarco Cannarsa; Khai T. Nguyen; Carlo Sinestrari. - In: MATHEMATISCHE ANNALEN. - ISSN 0025-5831. - STAMPA. - 356:(2013), pp. 23-43. [10.1007/s00208-012-0835-8]
Singular gradient flow of the distance function and homotopy equivalence
ALBANO, PAOLO;
2013
Abstract
Let M be a Riemannian manifold and let Ω be a bounded open subset of M. It is well known that significant information about the geometry of Ω is encoded into the properties of the distance, d∂Ω , from the boundary of Ω. Here, we show that the generalized gradient flow associated with the distance preserves singularities, that is, if x0 is a singular point of d∂Ω then the generalized characteristic starting at x0 stays singular for all times. As an application, we deduce that the singular set of d∂Ω has the same homotopy type as Ω.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
