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

TL;DR

This paper constructs a family of constant curvature one surfaces with two conical singularities on the 2-sphere, establishing a geometric correspondence with certain metrics and partially solving an open problem in the field.

Contribution

It provides an explicit construction of isometric immersions of metrics with two conical singularities into Euclidean 3-space, advancing understanding of the Gálvez-Hauswirth-Mira problem.

Findings

Constructed a family of constant curvature one surfaces with two conical singularities.

Established a geometric correspondence between metrics and immersions.

Provided a partial solution to the Gálvez-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).