Asymptotic behavior of flows by powers of the Gaussian curvature

Abstract

exists up to some maximal time, when the enclosed volume converges to zero.In the special case α=1/n, Chow [9] proved convergence to a round sphere.Moreover, Chow [10] obtained interesting Harnack inequalitites for flows, by powers of the Gaussian curvature (see also [17]).In the affine invariant case α=1/(n+2), Andrews [1] showed that the flow converges to an ellipsoid.This result can alternatively be derived from a theorem of Calabi [7], which asserts that the only self-similar solutions for α=1/(n+2) are ellipsoids.The arguments in [7] and [1] rely crucially on the affine invariance of the equation, and do not generalize to other exponents.In the special case of surfaces in R 3 (n=2), Andrews [2] proved that flow converges to a sphere when α=1; this was later extended in [4] to the case n=2 and α∈ 1 2 , 2 .The results in [2] and [4] rely on an application of the maximum principle to a suitably chosen function of the curvature eigenvalues; these techniques do not appear to work in higher dimensions.However, it is known that the flow converges to a self-similar solution for every n 2 and every α 1/(n+2).This was proved by Andrews [3] for α∈[1/(n+2), 1/n]; by Guan and Ni [16] for α=1; and by Andrews, Guan, and Ni [5] for all α∈(1/(n+2), ∞).One of the key ingredients in these results is a monotonicity formula for an entropy functional.This monotonicity was discovered by Firey [14] in the special case α=1.

Locations

• Acta Mathematica - View - PDF
• arXiv (Cornell University) - View - PDF
• Project Euclid (Cornell University) - View - PDF