The Qk flow on complete non-compact graphs

We establish the long time existence of complete non-compact weakly convex and smooth hypersurfaces Sigma(t) evolving by the Q(k)-flow. We show that the maximum existence time T depends on the dimension d(W) of the vector space W:={w is an element of Rn+1 : sup(X is an element of Sigma 0) vertical bar < X, w >vertical bar = +infinity} which contains each direction inwhich our initial data Sigma(0) is infinite. If d(W) = dim(W) >= n-k+ 1, then the solution Sigma(t) exists for all time t is an element of (0,+infinity); if d(W) = dim(W) <= n- k, then the solution Sigma(t) exsist up to some finite time T < +infinity. In the latter case, the trace at infinity Gamma(t) of the solution Sigma(t) is a closed convex viscosity solution of the (n - d(W))-dimensional Q(k) flow on t is an element of (0, T).

Automated summary

This paper establishes the long time existence of complete non-compact weakly convex and smooth hypersurfaces evolving by the Q(k)-flow. The maximum existence time is shown to depend on the dimension of a vector space, and the paper discusses the conditions for the existence of solutions in different dimensions.