[Paper Review] Rigidity of a family of spherical conical metrics

Xuwen Zhu|arXiv (Cornell University)|Feb 6, 2019

Geometric Analysis and Curvature Flows25 references4 citations

TL;DR

This paper establishes local geometric rigidity for a family of spherical conical metrics on the 2-sphere with four cone points, where one angle is $4 heta$ and the others are $\alpha$, $\beta$, and $\alpha+\beta$, with $\alpha, \beta, \alpha+\beta \notin 2\pi\mathbb{Z}$. Using synthetic geometry and spherical triangle analysis, it proves that any small perturbation of such a metric must be isometric to another metric in the same one-parameter family, implying that cone point positions are rigid when cone angles are fixed.

ABSTRACT

We study the deformation of spherical conical metrics with at least some of the cone angles larger than $2π$. We show in this note via synthetic geometry that for one family of such metrics, there is local rigidity in the choice of cone positions if angles are fixed. This gives an evidence of the analytic obstruction considered in recent works of Mazzeo and author.

Motivation & Objective

• To investigate the deformation space of spherical conical metrics with at least one cone angle exceeding $2\pi$.
• To determine whether the positions of cone points remain rigid under small perturbations when cone angles are fixed.
• To provide geometric evidence for analytic obstructions in the moduli space of such metrics, as suggested by prior work of Mazzeo and others.
• To establish that for a specific family of metrics with angles $ (\alpha, \beta, \alpha+\beta, 4\pi) $, the only nearby metrics are those in the one-parameter family obtained by gluing two spherical footballs.
• To demonstrate that the only admissible configuration under small perturbations is the symmetric gluing construction, using spherical triangle inequalities and extremal angle sum analysis.

Proposed method

• Decomposes the spherical surface into four geodesic triangles via a specific cut-and-glue decomposition along three geodesics.
• Defines 'triangulated metrics' as those with the same geodesic decomposition and uses a length-based parametrization of the metric space.
• Applies spherical trigonometry, particularly Napier’s analogies, to analyze angle sums in spherical triangles with fixed side lengths and opposite angles.
• Uses Lemma 3 to show that the sum of two angles in a spherical triangle is extremal (min or max) precisely when the triangle is isosceles, under fixed opposite side and angle.
• Analyzes the total angle sum around the $4\pi$ cone point by comparing configurations where triangle pairs are isosceles versus non-isosceles.
• Employs symmetry and extremal analysis to show that only the symmetric gluing of two footballs yields a total angle sum of exactly $4\pi$, ruling out asymmetric perturbations.

Experimental results

Research questions

• RQ1 Can small perturbations of a spherical conical metric with cone angles $ (\alpha, \beta, \alpha+\beta, 4\pi) $, where $\alpha, \beta, \alpha+\beta \notin 2\pi\mathbb{Z}$, result in a different cone point configuration while preserving the angles?
• RQ2 Is the geometric construction of gluing two spherical footballs the only possible realization of such a metric under small deformations?
• RQ3 Does the condition $\chi(M, \vec{\beta}) = 1$ and equality in Mondello-Panov’s inequality lead to rigidity in cone point positions?
• RQ4 What constraints do spherical triangle inequalities and extremal angle sum behavior impose on the possible configurations of nearby metrics?
• RQ5 Can synthetic geometry techniques alone establish rigidity in the moduli space of spherical conical metrics with large cone angles?

Key findings

• For any fixed $\vec{\beta} = (\alpha, \beta, \alpha+\beta, 4\pi)$ with $\alpha, \beta, \alpha+\beta \notin 2\pi\mathbb{Z}$, the only nearby triangulated metrics are isometric to $g_s$ for some $s$ in the one-parameter family.
• The only configuration achieving a total angle sum of exactly $4\pi$ at the $4\pi$ cone point is the symmetric gluing of two spherical footballs with angles $\alpha$ and $\beta$.
• When cone angles are fixed and $\alpha \neq \beta$, the only possible perturbation preserving the metric structure is the one-parameter family obtained by varying the length of the gluing slit.
• Any deviation from the symmetric configuration (e.g., $\ell_3 \neq \ell_4$) results in an angle sum strictly greater than or less than $4\pi$, violating the cone angle condition.
• The extremal angle sum in a spherical triangle with fixed opposite side and angle is achieved only in isosceles configurations, which constrains the possible metric deformations.