Non-sufficiency of smoothness in the gradient conjecture

TL;DR

This paper demonstrates that smoothness alone does not guarantee the gradient conjecture's conditions, providing a counterexample where a smooth function's gradient flow converges but secants do not have a limit.

Contribution

It presents a counterexample showing that smoothness without analyticity can fail the gradient conjecture's secant limit property.

Findings

Counterexample for smooth functions where secants do not converge

Smooth functions can satisfy strong convergence properties

Analyticity is essential for the gradient conjecture's secant limit

Abstract

It is well known that for analytic cost functions, gradient flow trajectories have finite length and converge to a single critical point. The gradient conjecture of R. Thom states that, again for analytic cost functions, whenever the gradient flow trajectory converges, the limit of its unit secants exists. One might think that already the convergence of the gradient flow trajectory to a critical point is enough to ensure that the unit secants have a limit, but this does not hold in general - the gradient conjecture is to a certain extend sharp. We provide a counterexample in case of the missing analyticity assumption, that is a smooth (non-analytic) cost function $f$, where the limit of unit secants does not exist. In addition, $f$ satisfies even a strong geometric length-distance convergence property.