Concentration-compactness for the geometric polyharmonic heat flow

Abstract

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 \(\R^3\) as the model case. The flow is the \((2p+2)\)-order normal evolution \[ \partial_t f=(-1)^{p+1}\Delta^p H\,\nu,\qquad p\geq1, \] 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 \(\varepsilon\)-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 \(\|A^o\|_2^2<\varepsilon\), where \(\varepsilon>0\) depends only on the order of the flow, the solution exists for all time and converges exponentially in \(C^\infty\) to a round sphere with the preserved enclosed volume.