Representation formulae for higher order curvature flows

In [25], Smoczyk showed that expansion of convex curves and hypersurfaces by the reciprocal of the harmonic mean curvature gives rise to a linearsecond order equation for the evolution of the support function, with corresponding representation formulae for solutions. In this article we consider L-2(d theta)-gradient flows for a class of higher-order curvature functionals. These give rise to higher order linear parabolic equations for which we derive similar representation formulae for their solutions. Solutions exist for all time under natural conditions on the initial curve and converge exponentially fast in the smooth topology to multiply-covered circles. We consider both closed, embedded convex curves and closed, convex curves of higher rotation number. We give some corresponding remarks where relevant on open convex curves. We also consider corresponding `globally constrained' flows which preserve the length or enclosed area of the evolving curve and a higher order approach to the Yau problem of evolving one convex planar curve to another. In an Appendix, we give some related second order results, including a version of the Yau problem for star-shaped curves.

Automated summary

In this article, we studied L-2(d theta)-gradient flows for higher-order curvature functionals and derived similar representation formulae for their solutions. Solutions exist for all time under natural conditions on the initial curve and converge exponentially fast in the smooth topology to multiply-covered circles. We considered different types of curves and also discussed corresponding `globally constrained' flows and a higher order approach to the Yau problem.