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.
Paolo Albano, Piermarco Cannarsa, Khai T. Nguyen, Carlo Sinestrari (2013). Singular gradient flow of the distance function and homotopy equivalence. MATHEMATISCHE ANNALEN, 356, 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.
