Entire self-expanders for power of $\sigma_k$ curvature flow in Minkowski space

TL;DR

This paper establishes the existence of entire self-expanders for the power of k curvature flow in Minkowski space, extending the understanding of such flows beyond previous results which focused on convergence and bounded curvature.

Contribution

The paper proves the existence of entire self-expanders for the k curvature flow in Minkowski space, a problem not previously addressed in the literature.

Findings

Existence of entire self-expanders for k curvature flow in Minkowski space.

Self-expanders satisfy a specific k curvature equation involving Minkowski inner product.

The results extend the theory of curvature flows to new geometric settings.

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.