Gradient Vector Fields Do Not Generate Twister Dynamics

Abstract

Gradient Conjecture states that a solution g of an analytic gradient vector field X approaching to a singularity P of X has a tangent at P. A stronger version asserts that g does not meet an analytic hypersurface an infinite number of times (it is non-oscillating). We prove, in dimension 3, that if g is ``infinitely near'' an analytic curve G not composed of singularities of X, then g is non-oscillating and, moreover, it does not spiral around G in a precise sense.