An Interpolation from Sol to Hyperbolic Space

Abstract

We study a one-parameter family of nonisomorphic solvable Lie groups, which, when equipped with canonical left-invariant metrics, $$d{s}^2=e^{-2z}d{x}^2+e^{2\alpha z}d{y}^2+d{z}^2$$ becomes an interpolation from a model of the Sol geometry to a model of Hyperbolic Space, with a stop at ${\mathbb{H}}^2\times \mathbb{R}$. These Lie groups are also Bianchi groups of Type VI with orthogonal coordinates. As a continuation of joint work with Richard Schwartz on Sol, we primarily analyze those Lie groups in our interpolation with some positive sectional curvature. Our main result is a characterization of the cut locus at the identity of the group that maximizes scalar curvature.

50 pages. We improved the exposition, added figures, and included some numerical data. Otherwise, the proofs are the same

References in corpus (2)

• Families of Geometries, Real Algebras, and Transitions
• On Area Growth in Sol

Cited by in corpus (2)

• Rotating black holes with Nil or SL(2,$\mathbb{R}$) horizons
• Some New Results in Geometric Analysis