Computer-assisted construction of SU(2)-invariant negative Einstein metrics

Area of Science:

• Differential Geometry
• Mathematical Physics

Background:

• Einstein metrics are fundamental in general relativity and differential geometry.
• Constructing complete Einstein metrics, especially with negative curvature, is a challenging problem.

Purpose of the Study:

• To construct a new family of triaxial SU(2)-invariant complete negative Einstein metrics.
• To investigate properties such as conformal compactness and non-Kähler nature.

Main Methods:

• Utilizing rigorous numerics to approximate an Einstein metric in a critical region.
• Employing fixed-point methods to perturb the approximate metric into a genuine Einstein metric.
• Analyzing the metric's behavior near the boundary to establish asymptotic hyperbolicity.

Main Results:

• A 2-parameter family of new triaxial SU(2)-invariant complete negative Einstein metrics on O(-4) over CP^1.
• Demonstration that these metrics are conformally compact, non-Kähler, and non-self-dual.
• Proof of extension to complete, asymptotically hyperbolic Einstein metrics.

Conclusions:

• The study successfully constructs novel Einstein metrics with specific properties.
• The methods provide a framework for constructing and analyzing complex geometric structures.
• The findings contribute to the understanding of negative Einstein metrics and their global properties.