The $Q_k$ flow on complete non-compact graphs

Abstract

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 $W{:}{=}\{w \in \mathbb {R}^{n+1}: \sup _{X\in \Sigma _0} |\langle X,w\rangle | = +\infty \}$ which contains each direction in which our initial data $\Sigma _0$ is infinite. If $d_W=\text {dim}(W) \ge n-k+1$, then the solution $\Sigma _t$ exists for all time $t \in (0,+\infty )$; if $d_W=\text {dim}(W) \le n-k$, then the solution $\Sigma _t$ exsist up to some finite time $T < +\infty$. 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 $t \in (0,T)$.