| We study the long-time convergence of harmonic map heat flows from a closed Riemann surface into a compact Riemannian manifold. P. Topping constructed an example of a flow that does not converge in the infinite-time limit. Motivated by the observation that Topping's flow has accumulation points at which the Hessian of the energy function is degenerate, we prove convergence under the assumptions that (a) the Hessian of the energy at an accumulation point is positive definite, and (b) no bubbling occurs at infinite time. In addition, we present examples of heat flows for geodesics which show that the convexity of the energy function and convergence as t → infinity may not hold even for 1-dimensional harmonic map heat flows. |
