Isometric immersions of constant curvature one metrics on the 2-sphere with two conical singularities into Euclidean 3-space: A partial solution to the Gálvez-Hauswirth-Mira problem

2025/02/19 by Zhiqiang Wei, Wei, Zhiqiang
Mathematics · #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2502.13748

Abstract

In this paper, we establish a geometric correspondence between constant curvature one metrics with two conical singularities on S² and isometric immersions into Euclidean 3-space 𝔼³. 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 Gálvez, Hauswirth, and Mira (Adv in Math. 241(2013) 103-126).

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