Representation formulae for higher-order linear curvature flows

Abstract In Smoczyk (Elem Math 60(2):57-65, 2005. https://doi.org/10.4171/EM/9 ) showed that the expansion of convex hypersurfaces by the reciprocal of the harmonic mean curvature gives rise to a linear second-order equation for the evolution of the support function, with corresponding representation formulae for solutions. In this article, we consider $$L^2(d\sigma )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>d</mml:mi> <mml:mi>σ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> -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. With suitable restrictions on the initial convex hypersurface, solutions exist for all time and converge exponentially fast in the smooth topology to spheres. We also consider evolution by similar flows that keep certain integrals fixed and those that evolve one convex hypersurface to another. In an Appendix, we give some related second-order results.