The Minkowski problem in the sphere

Abstract

$\def\S{\mathbb{S}} \def\M{\mathcal{M}}$ Given a function $f \gt 0$ on the unit sphere $\S^{n+1}$, the Minkowski problem in the sphere concerns the existence of convex hypersurfaces $\M \subset \S^{n+1}$ such that the Gauss curvature of $\M$ at $z$ is equal to $f(\nu(z))$, where $\nu(z)$ is the unit outer normal of $\M$ at $z$. We use the min‑max principle and the Gauss curvature flow to prove that there are at least two solutions to the problem. By using the rotating plane method in the sphere, we also show the existence of a rotationally symmetric and monotone function $f$ such that there are exactly two solutions.