Slowly converging geometric equations
We discuss slowly converging solutions arising from the study of isolated singularities in geometric variational problems such as minimal surface, mean curvature flow, Harmonic map and its heat flow. They have previously been constructed assuming the Adams-Simon positivity condition. We identify a necessary condition for slowly converging solutions to exist, which we refer to as the Adams-Simon non-negativity condition. Additionally, we characterize the rate and direction of convergence for these solutions. Our result partially confirms Thom's gradient conjecture in the context of infinite-dimensional problems. This is a joint work with Pei-Ken Hung at Minnesota.