For any 1 ≤ k ≤ n, the k-Hessian curvature flow with prescribed gradient image on the boundary has a smooth solution for all time that converges to a translating solution.
Andrews, Gauss curvature flow: the fate of the rolling stones, Invent
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Second boundary value problem for the Hessian curvature flow
For any 1 ≤ k ≤ n, the k-Hessian curvature flow with prescribed gradient image on the boundary has a smooth solution for all time that converges to a translating solution.