Entire self-expanders for power of ${\sigma }_k$ curvature flow in Minkowski space

Abstract

In [19], we prove that if an entire, spacelike, convex hypersurface ${\mathcal{M}}_{u_0}$ has bounded principal curvatures, then the ${\sigma }_k^{1/\alpha }$ (power of ${\sigma }_k$) curvature flow starting from ${\mathcal{M}}_{u_0}$ admits a smooth convex solution $u$ for $t>0.$ Moreover, after rescaling, the flow converges to a convex self-expander $\tilde{\mathcal{M}}=\{(x,\tilde{u}(x))\mid x\in {\mathbb{R}}^n\}$ that satisfies ${\sigma }_k(\kappa [\tilde{\mathcal{M}}])=(-⟨X_0,{\nu }_0⟩{)}^{\alpha }.$ Unfortunately, the existence of self-expander for power of ${\sigma }_k$ curvature flow in Minkowski space has not been studied before. In this paper, we fill the gap.