Evolution of convex hypersurfaces by a fully nonlinear flow. (January 2016)

• Record Type: Journal Article
• Authors: Liu, Yan-nan; Ju, Hong-jie
• Is Part Of: Nonlinear analysis. Volume 130(2015)
• Page Start: 47
• Page End: 58
• Publication Date: 2016-01
• Subjects: 35K45 -- 53A05; Parabolic equation -- Fully nonlinear curvature flow -- Forced term; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.09.014
• Languages: English
• ISSNs: 0362-546X

Abstract

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.