A Class of inverse curvature type flows in Rn+1

Abstract In this paper, we consider an expanding flow of smooth, closed, (, k) -convex hypersurfaces in Euclidean R^n+1 with speed u^^ₖ^-{k} ( () ), where u, are the support function and radical function of the hypersurface, respectively, , ¹, >0, k is an integer and 1 k n, =Hg-h, the first Newton transformation of the second fundamental form h, () denote the eigenvalues of g^-1. For ++ 1, we prove that the flow has a unique smooth and (, k) -convex solution for all time, and converges smoothly after normalisation, to a sphere centered at the origin. Moreover, for ++< 1, we prove that the flow with the speed fu^^ₖ^-{k} ( () ) exists for all time, and converges smoothly after normalisation to a soliton, which is a solution of fu^-1^ₖ^-{k} ( () ) =, provided that f is a smooth positive function on Sⁿ and is a positive constant. What's more, we can use a more general flow to prove the existence of solution to a class of Hessian quotient equations again.