existence and uniqueness of solutions to curvature problems

This research project is focused on understanding of curvature problems. Curvature is a measure of how much a surface deviates from being flat and is an important concept in geometry and physics. The project has three main parts. The first part of the project will involve extending recent work on a specific type of curvature problem called the Christoel-Minkowski problem to include more complex cases, such as anisotropic (direction-dependent) curvature and mixed curvature problems. This will require further exploration of new mathematical ideas that have been developed in previous research on this topic. The second part of the project will focus on studying the stability of a specific type of curvature function, which is constant only for origin-centred ellipsoids. Finally, the third part of the project will involve extending a new proof of a theorem called the constant rank theorem, which has been developed by the investigator and their collaborators to include cases where the surface has a "free boundary." This will allow the investigator to study curvature problems in a broader range of situations. Overall, this research project aims to deepen our understanding of a large class of curvature problems, which have important applications in fields such as physics and engineering.