Evolution of convex hypersurfaces by a fully nonlinear flow. (January 2016) - Liu, Yan-nan; Ju, Hong-jie
Abstract: In this paper, we study the evolution of convex hypersurfaces by a fully nonlinear function of curvature minus an external force field c . We prove that the flow will preserve the convexity for any c . When c < g on the initial surface, where g is the fully nonlinear function, we prove that the flow will expand the hypersurface for all time. If on initial surface M 0 the minimal principal radius of curvature r min satisfies r min > n c and minimum of the support function s satisfies s min > n c, then after a scaling the hypersurface will converge to a sphere. If c > g on the initial surface M 0 and the diameter of M 0 satisfies diam ( M 0 ) < 2 n c, we show that the maximal existence time of the flow is finite.