Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Zhiqiang Wei

TL;DR

This work constructs an explicit correspondence between constant curvature one metrics with two conical singularities on $S^{2}$ and isometric immersions into $\mathbb{E}^{3}$, yielding a partial solution to the open problem of realizing such metrics as $K=1$ immersions. By leveraging Troyanov’s classification and the football metric framework, the authors derive a concrete, parameterized immersion family with $\lambda B=\alpha$ and provide explicit formulas for the embedding when two equal-angle singularities are present. The main contribution is an explicit immersion $\vec r$ with $K=1$ that is smooth away from the singularities, including a rotationally symmetric derivation and a suite of explicit examples illustrating various conical angle configurations. The results broaden understanding of the intrinsic-extrinsic correspondence for spherical metrics with conical singularities and offer practical geometric realizations, including non-embedded and branched-cover constructions, reinforcing the connection between conformal geometry and isometric immersion theory.

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 Gálvez, Hauswirth, and Mira (Adv in Math. 241(2013) 103-126).

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

TL;DR

This work constructs an explicit correspondence between constant curvature one metrics with two conical singularities on and isometric immersions into , yielding a partial solution to the open problem of realizing such metrics as immersions. By leveraging Troyanov’s classification and the football metric framework, the authors derive a concrete, parameterized immersion family with and provide explicit formulas for the embedding when two equal-angle singularities are present. The main contribution is an explicit immersion with that is smooth away from the singularities, including a rotationally symmetric derivation and a suite of explicit examples illustrating various conical angle configurations. The results broaden understanding of the intrinsic-extrinsic correspondence for spherical metrics with conical singularities and offer practical geometric realizations, including non-embedded and branched-cover constructions, reinforcing the connection between conformal geometry and isometric immersion theory.

Abstract

In this paper, we establish a geometric correspondence between constant curvature one metrics with two conical singularities on and isometric immersions into Euclidean 3-space . 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).

Paper Structure

This paper contains 5 sections, 2 theorems, 55 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Proof of Theorem \ref{['Mthm']}
  4. Derivation of the map $\vec{r}$ in Section 3
  5. Some explicit examples

Key Result

lemma 1

Let ${\rm ds^{2}}$ be a football metric on $S^{2}$ with conical angles $2\pi\alpha$ and $2\pi\alpha$. Then, under the geodesic coordinates, ${\rm ds^{2}}$ can be expressed as where $(r,\theta)\in(0,\pi)\times[0,2\pi)$.

Figures (6)

  • Figure 1: Football of angles $\pi$ and $\pi$.
  • Figure 2: Football of angles $2\pi$ and $2\pi$.
  • Figure 3: Football of angles $4\pi$ and $4\pi$.
  • Figure 4: Football of angles $4\pi$ and $4\pi$.
  • Figure 5: Football of angles $4\pi$ and $4\pi$.
  • ...and 1 more figures

Theorems & Definitions (7)

  • remark 1
  • definition 1
  • definition 2
  • lemma 1
  • proposition 1
  • proof
  • remark 2