  R. Thom's gradient conjecture states that if a gradient flow of an analytic
function converges to a limit, it does so along a unique limiting direction. In
this paper, we extend and settle this conjecture in the context of infinite
dimensional problems. Building on the foundational works of {\L}ojasiewicz, L.
Simon, and the resolution of the conjecture for finite dimensional cases by
Kurdyka-Mostowski-Parusinski, we focus on nonlinear evolutions on Riemannian
manifolds as studied by L. Simon. This framework includes geometric PDEs such
as minimal surface, harmonic map, mean curvature flow, and normalized Yamabe
flow. Our main result not only confirms the uniqueness of the limiting
direction but also characterizes the rate of convergence and possible limiting
directions for both classical and infinite dimensional settings.
