2016/10/27 by Simon Brendle, Kyeongsu Choi, Brendle, Simon +3 · 13 citations
Mathematics · #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities #Geometry and complex manifolds
paper · pdf · doi:10.4310/acta.2017.v219.n1.a1
exists up to some maximal time, when the enclosed volume converges to zero. In the special case =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 In the affine invariant case =1/(n+2), Andrews This result can alternatively be derived from a theorem of Calabi 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 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 1/(n+2). This was proved by Andrews [3] for [1/(n+2), 1/n]; by Guan and Ni [16] for =1; and by Andrews, Guan, and Ni [5] for all (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 =1.