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On restricted analytic gradients on analytic isolated surface singularities Grandjean, Vincent Sanz Sánchez, Fernando Gradient vector field Oscillating trajectories Blowing-up Reduction of singularities Let (X,0) be a real analytic isolated surface singularity at the origin 0 of Rn and let g be a real analytic Riemannian metric at 0∈Rn. Given a real analytic function f0:(Rn,0)→(R,0) singular at 0, weprove that the gradient trajectories for the metric g|X\0 of the restriction (f0|X) escaping from or ending up at 0 do not oscillate. Such a trajectory is thus a sub-pfaffian set. Moreover, in each connected component of X\0 where the restricted gradient does not vanish, there is always a trajectory accumulating at 0 and admitting a formal asymptotic expansion at 0. ThesecondauthorwaspartiallysupportedbytheresearchprojectsVA059A07(JuntadeCastillayLeón)andMTM2007-66262(MinisteriodeEducaciónyCiencia)andbyPlanNacionaldeMovilidaddeRR.HH.2008/11,Modalidad“JoséCastillejo” 2024-06-21T10:24:15Z 2024-06-21T10:24:15Z 2013 info:eu-repo/semantics/article info:eu-repo/semantics/submittedVersion J. DifferentialEquations255(2013)1684–1708 0022-0396 https://uvadoc.uva.es/handle/10324/68177 10.1016/j.jde.2013.05.020 1684 7 1708 Journal of Differential Equations 255 eng info:eu-repo/semantics/restrictedAccess Elsevier application/pdf Elsevier https://uvadoc.uva.es/bitstream/10324/68177/3/2013_Grandjean-Sanz_JDEQ.pdf.jpg Hispana TEXT http://rightsstatements.org/vocab/CNE/1.0/ UVaDOC. Repositorio Documental de la Universidad de Valladolid https://uvadoc.uva.es/handle/10324/68177