Arbitrarily high order concentration-compactness for curvature flow

We extend Struwe and Kuwert-Schaetzle's concentration-compactness method for analyzing geometric evolution equations to flows of an arbitrarily high order, with the geometric polyharmonic heat flow (GPHF) of surfaces, a generalization of surface diffusion flow, as an exemplar. For the (GPHF) we apply the technique to deduce localized energy and interior estimates, a concentration-compactness alternative, pointwise curvature estimates, a gap theorem, and study the blowup at a singular time. This gives general information on the behavior of the flow for any initial data. Applying this for initial data satisfying $||A^o||_2^2 < \varepsilon$ where $\varepsilon$ is a universal constant, we perform global analysis to obtain exponentially fast full convergence of the flow in the smooth topology to a standard round sphere. This is joint work with James McCoy, Scott Parkins, and Valentina-Mira Wheeler.