Thom's gradient conjecture for nonlinear evolution equations

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.