On restricted analytic gradients on analytic isolated surface singularities

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:255

Article

Grandjean, Vincent¹  Sanz, Fernando² 

[1] Univ Fed Ceara, Dept Matemat, BR-60455760 Fortaleza, Ceara, Brazil

[2] Univ Valladolid, Fac Ciencias, Dept Algebra Anal Matemat Geometria & Topol, E-47006 Valladolid, Spain

关键词: Gradient vector field;    Oscillating trajectories;    Blowing-up;    Reduction of singularities;   

DOI  :  10.1016/j.jde.2013.05.020

来源: Elsevier

【 摘 要 】

Let (X, 0) be a real analytic isolated surface singularity at the origin 0 of R-n and let g be a real analytic Riemannian metric at 0 is an element of R-n. Given a real analytic function f(0) : (R-n, 0) -> (R, 0) singular at 0, we prove that the gradient trajectories for the metric g vertical bar(X\0) of the restriction (f(0)vertical bar(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. (c) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

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