On restricted analytic gradients on analytic isolated surface singularities

Abstract

Abstract. Let (X, 0) be a real analytic isolated surface singularity at the origin 0 of a real analytic manifold (R n , 0) equipped with a real analytic metric g. Given a real analytic function f0 : (R n , 0) → (R, 0) singular at 0, we prove that the gradient trajectories for the metric g| X\0 of the restriction (f0|X ) escaping from or ending up at the origin 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.

Gradient Trajectories For Plane Singular Metrics I: Oscillating Trajectories

“…In particular this naive point of view is forgetting that in this description the degeneracy of the metric is forced by the space, since the restricted metric can extend to the whole ambient space as a standard Riemannian metric. Consequently The singular metric of [5] comes from the restriction of a Riemannian metric to a singular "cone". Note that the asymptotic behaviour at the singular point of this restricted metric is just the restriction of the ambient metric to the asymptotic behaviour of the singular "cone" at its tip, namely the limits at the tip of the tangent spaces to the surface.…”

On Hessian Limit Directions along Gradient Trajectories

“…We recall that plane gradient trajectories do not oscillate at their limit point, and that in higher dimension non-oscillating gradient trajectories exist as well [12,13,4,6].…”

“…Whether any gradient trajectory is oscillating or not at its limit point is a very hard problem in general, which is not understood beyond the special cases dealt with in [5,8,15].…”

“…The plane smooth case is very well understood, and (bounded) plane gradients curves are definable in the Pfaffian closure of R an . But a few special cases trying to address the non-spiraling/non-oscillating behavior of such trajectories [13,4,6], and the main result of [12], that is all that is known. The aim of this note is to report an intriguing and not expected result, along gradient trajectories that are non-oscillating at their limit point.…”

A convex function satisfying the Lojasiewicz inequality but failing the gradient conjecture both at zero and infinity

“…For decades, this has been known as the (Thom) gradient conjecture, see [1,29]. (For the more general problem of non-oscillation of trajectories, we refer to [4,12,24].) The gradient conjecture makes sense for any gradient dynamics for which bounded orbits converge.…”