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

TL;DR

This paper constructs a smooth convex function that satisfies the Lojasiewicz inequality at zero but exhibits spiraling gradient orbits at both zero and infinity, thereby failing Thom's gradient conjecture in these regions.

Contribution

It provides a counterexample demonstrating that the Lojasiewicz inequality does not imply the gradient conjecture for certain convex functions.

Findings

The constructed function satisfies the Lojasiewicz inequality at zero.

Gradient orbits spiral around zero and at infinity.

Thom's gradient conjecture fails at both zero and infinity for this function.

Abstract

We construct an example of a smooth convex function on the plane with a strict minimum at zero, which is real analytic except at zero, for which Thom's gradient conjecture fails both at zero and infinity. More precisely, the gradient orbits of the function spiral around zero and at infinity. Besides, the function satisfies the Lojasiewicz gradient inequality at zero.