Evolution of convex entire graphs by curvature flows

Abstract

Abstract:We consider the evolution of an entire convex graph in euclidean space with speed given by a symmetric function of the principal curvatures. Under suitable assumptions on the speed and on the initial data, we prove that the solution exists for all times and it remains a graph. In addition, after appropriate rescaling, it converges to a homothetically expanding solution of the flow. In this way, we extend to a class of nonlinear speeds the well known results of Ecker and Huisken for the mean curvature flow. MSC: 53C44

$H^α$-flow of mean convex, complete graphical hypersurfaces

“…Concerning entire graphs, we have already mentioned [8], where the H α -flow is considered, too, and the results of which we could improve by dropping a certain condition called "ν-condition" as well as expanding the result to complete graphs. Still concerning entire graphs, the author wants to mention [9], where flows with speed S 1/k k with S k an elementary symmetric polynomial of the principal curvatures are considered, and [1], where general curvature flows of homogeneity one are investigated. Inspired by the mean curvature flow of complete graphs there is a number of papers that are concerned with the flow of complete graphs by various normal speeds.…”

Evolution of complete noncompact graphs by powers of curvature function

“…Under the weak convexity assumption, Choi and Daskalopoulos [15] proved the long time existence of complete convex solution for E k E k−1 -flow. Recently, Alessandroni and Sinestrari [1] considered the evolution of entire convex graph by a general symmetric function F of principal curvatures. If velocity F is concave and inverse concave, they proved the solution exists for all times provided F ≥ εH holds for some positive constant ε.…”