A Class Of Curvature Flows Expanded By Support Function And Curvature Function

In this paper,we first consider an expanding flow of closed,smooth,uniformly convex hypersurface in Euclidean Rn+1 with speed u~?f~?(?,??R1),where u is support function of the hypersurface,f is a smooth,symmetric,homogenous of degree one,positive function of the principal curvature radii of the hypersurface.If ??0<??1-?,we prove that the flow has a unique smooth and uniformly convex solution for all time,and converges smoothly after normalization,to a round sphere centered at the origin.After that,we improve the method.We consider a class of expanding flows of closed,smooth,star-shaped hypersurface in Euclidean space Rn+1 with speed u~?f-?,where u is the support function of the hypersurface,f is a smooth,symmetric,homogenous of degree one,positive function of the principal curvatures of the hypersurface on a convex cone.For ??0<??1-?,we prove that the flow has a unique smooth solution for all time,and converges smoothly after normalization,to a sphere centered at the origin.In particular,the results of Gerhardt[19]and Urbas[42]can be recovered by putting?=0 and ?=1 in our first result.If the initial hypersurface is convex,this is our previous work[14].Finally,we find a family of monotone quantities along the flows in Rn+1.As applications,we give a proof of a family of inequalities involving the weighted integral of kth elementary symmetric function for k-convex,star-shaped hypersurfaces,which is an extension of the quermassintegral inequalities in[23].