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Entire self-expanders for power of $σ_k$ curvature flow in Minkowski space (2205.06853v1)

Published 13 May 2022 in math.DG

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 $\sigma_k(\kappa[\tilde{\mathcal{M}}])=(-\left<X_0, \nu_0\right>)\alpha.$ 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.

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