Isometric immersions of constant curvature one metrics on the 2-sphere with two conical singularities into Euclidean 3-space: A partial solution to the G\'{a}lvez-Hauswirth-Mira problem

Abstract

In this paper, we establish a geometric correspondence between constant curvature one metrics with two conical singularities on $S^{2}$ and isometric immersions into Euclidean 3-space $\mathbb{E}^{3}$. Specifically, we explicitly construct a family of surfaces with constant curvature one, each of which is endowed with two conical singularities. This construction provides a partial solution to an open problem proposed by G\'{a}lvez, Hauswirth, and Mira (Adv in Math. 241(2013) 103-126).