Concentration-compactness for the geometric polyharmonic heat flow

We develop a concentration-compactness theory for geometric evolution equations of arbitrarily high order, using the geometric polyharmonic heat flow of closed immersed surfaces in ${\mathbb{R}}^3$ as the model case. The flow is the $(2p+2)$-order normal evolution $${\mathrm{\partial }}_{t}f=(-1{)}^{p+1}{\mathrm{\Delta }}^{p}H\nu ,p\ge 1,$$ which includes the surface diffusion flow when $p=1$. We prove localised energy estimates with sharp cut-off bookkeeping, interior estimates, a lifespan/concentration alternative, tracefree-curvature $\epsilon$-regularity estimates, and a gap theorem for stationary solutions. These tools are then combined with a blowup argument, the preservation of signed enclosed volume, and the monotonicity of area to rule out singularities below a small tracefree-curvature threshold. Consequently, for connected initial immersions satisfying $\mathrm{\parallel }{A}^o{\mathrm{\parallel }}_2^2<\epsilon$, where $\epsilon >0$ depends only on the order of the flow, the solution exists for all time and converges exponentially in $C^{\mathrm{\infty }}$ to a round sphere with the preserved enclosed volume.