Explicit Isometric Immersions from Footballs to R3

In general, we call a one-dimensional singular non-CSC extremal Kähler metric HCMU metric. If an HCMU metric on S 2 just has two singularities, we call this kind of HCMU metrics football. Peng and Wu (Results Math 75:133, 2020) proved that any HCMU metric can be locally isometrically imbedded into R 3 as a Weingarten surface. In this paper, we will give a family of local explicit isometric immersions from a football to R 3 and prove that there exists a global isometric immersion from a football to R 3 .