On the gradient conjecture at infinity: Finite spherical length for gradient trajectories of definable potentials
We prove that for every open $ U \subseteq \mathbb R^n $ the spherical projection of every gradient trajectory $θ(t)$, $t \geq 0$, of a definable $C^1$-potential $F \colon U \to \mathbb R$ has finite length and, in particular, converges (directional convergence). In other words, every gradient trajectory of a definable $C^1$-potential has finite total angular motion. Here, definable means definable in an arbitrary o-minimal structure expanding the real field....
We prove that for every open $ U \subseteq \mathbb R^n $ the spherical projection of every gradient trajectory $θ(t)$, $t \geq 0$, of a definable $C^1$-potential $F \colon U \to \mathbb R$ has finite length and, in particular, converges (directional convergence). In other words, every gradient trajectory of a definable $C^1$-potential has finite total angular motion. Here, definable means definable in an arbitrary o-minimal structure expanding the real field....