# Convergence of Gauss curvature flows to translating solitons

## TL;DR

This paper studies the long-term behavior of the alpha-Gauss curvature flow for convex hypersurfaces, proving convergence to a unique translating soliton determined by the initial shape's asymptotic cylinder.

## Contribution

It establishes the convergence of the alpha-Gauss curvature flow to a unique translating soliton for non-compact convex hypersurfaces with specific initial conditions.

## Key findings

- Flow converges to a translating soliton as time approaches infinity.
- The limiting soliton is uniquely determined by the initial hypersurface's asymptotic cylinder.
- Results apply for alpha greater than 1/2 with initial data contained in a bounded cylinder.

## Abstract

We address the asymptotic behavior of the $\alpha$-Gauss curvature flow, for $\alpha >1/2$, with initial data a complete non-compact convex hypersurface which is contained in a cylinder of bounded cross section. We show that the flow converges, as $t \to +\infty$, locally smoothly to a translating soliton which is uniquely determined by the asymptotic cylinder of the initial hypersurface.