Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems (Q2302851)

Summary: In this paper we study a contracting flow of closed, convex hypersurfaces in the Euclidean space \(\mathbb R^{n+1}\) with speed \(f r^{\alpha} K\), where \(K\) is the Gauss curvature, \(r\) is the distance from the hypersurface to the origin, and \(f\) is a positive and smooth function. If \(\alpha \geq n+1\), we prove that the flow exists for all time and converges smoothly after normalisation to a soliton, which is a sphere if \(f \equiv 1\). Our argument provides a new proof in the smooth category for the classical Aleksandrov problem, and resolves the dual \(q\)-Minkowski problem introduced by \textit{Y. Huang} et al. [Acta Math. 216, No. 2, 325--388 (2016; Zbl 1372.52007)] for \(q < 0\). If \(\alpha < n+1\), corresponding to the case \(q > 0\), we also establish the same results for even functions \(f\) and origin-symmetric initial condition, but for \(f\) non-symmetric, a counterexample is given for the above smooth convergence.