  In [8], the gradient conjecture of R. Thom was proven for gradient flows of
analytic functions on $\mathbb{R}^n$. This result means that the secant at a
limit point converges, so that the flow cannot spiral forever. Once the
trajectory becomes sufficiently close to a critical point, the flow becomes a
simple scaling. Their paper is also significant in the number of auxiliary
results they prove about the convergence behaviour of gradient flows, on the
way to proving their main result. Many gradient flows of interest occur on
infinite-dimensional function spaces. And of considerable research interest
today are geometric flows with a gauge or diffeomorphism symmetry. We show that
the corresponding gradient conjecture holds also for parabolic flows on Hilbert
spaces, and for the extensively studied Yang-Mills flow.
