On restricted Analytic Gradients on Analytic Isolated Surface Singularities

TL;DR

This paper proves that gradient trajectories on real analytic isolated surface singularities do not oscillate and are sub-pfaffian, with trajectories near the singularity admitting formal asymptotic expansions, advancing understanding of singularity behavior.

Contribution

It establishes non-oscillation of gradient trajectories and their sub-pfaffian nature on real analytic isolated surface singularities, including asymptotic expansions.

Findings

Gradient trajectories do not oscillate near the singularity.

Trajectories are sub-pfaffian sets.

Existence of trajectories with formal asymptotic expansions.

Abstract

Let (X,O) be a real analytic isolated surface singularity at the origin o of a real analytic manifold M equipped with a real analytic metric g. Given a real analytic function f:(M,O) --> (R,0) singular at O, we prove that the gradient trajectories for the metric g|(X,O) of the restriction f|X escaping from or ending up at the origin O do not oscillate. Such a trajectory is thus a sub-pfaffian set. Moreover, in each connected component of X\O where the restricted gradient does not vanish, there is always a trajectory accumulating at O and admitting a formal asymptotic expansion at O