Boundary Value Problems Related to Extremal Kaehler Metrics

We will present a certain type of Riemannian metrics on the unit sphere. These metrics can each be defined via a smooth positive function of one variable satisfying certain boundary conditions. Inspired by the so called weighted extremal Kaehler metrics in higher dimensions (defined recently in the works by Apostolov, Calderbank, Gauduchon, Legendre, Maschler, and Lahdili), we then look at metrics whose defining functions solve a certain second order ordinary differential equation. In the end we relate our findings to Gaussian curvature and Einstein metrics, as well as the Calabi Functional.