Complete noncompact self-similar solutions of Gauss curvature flows. I: Positive powers (Q1392477)

In this well-written paper, the author uses techniques he developed in his earlier work on Monge-Ampére equations to classify all complete noncompact embedded convex hypersurfaces \(M\), moving by positive powers of their Gauss curvature \(K^\alpha \), either by homothety or by translation. It is shown that, with the exception of the hyperplane, each homothetic solution is enclosed in some convex cone \(C\) with vertex at the origin, such that \(M\) is asymptotic to \(\partial C\). Conversely, given any closed convex cone \(C\), there exists \(M\) as above, a homothetic solution to the \(K^\alpha\)-flow. The question of whether or not the hypersurface meets \(\partial C\) is examined and results depending on the geometry of \(C\) and/or values of \(\alpha\) are obtained. In the case of translating solutions, the hypersurface is essentially the union of a graph of a convex function on some convex \(\Omega \subset \mathbb{R}^n\) and some portion of the boundary of the cylinder \(\Omega \times \mathbb{R}\). The existence of translating solutions for every given speed is discussed.