# Hyperbolic

### Basic setup

Hyperbolic plane: H is given by $$z^2=x^2+y^2+1$$ (with z ≥ 1) (R3 with Minkowski metric)

Or H is the subset of $$\mathfrak{sl}(2,\,\mathbb{R})$$ defined by determinant = 1

Symmetry group: G = SL(2, R)

Phase space M = H x H x . . . x H (n copies)

Momentum map: J(A1, . . . An) = Σ κj Aj.

Momentum isotropy:

 μ Gμ 0 SL(2, R) det μ < 0 R det μ = 0 R det μ > 0 SO(2)

### All κi > 0

• then det(μ) > 0 and Gμ = SO(2).
• also J-1(μ) is bounded:
• use z-components of points: Σj κj zj is fixed (= z(μ)). So all zj < z(μ)/min(κi)

### 2 vortices

• For each μ, J-1(μ) is 4 - 3 = 1-dimensional, so (if connected) it must concide with the Gμ-orbit. In particular, this means reduced spaces are points. (But how is reduction done for non-compact group?)
• μ≠0 (unless A1=A2 and κ1=-κ2)
• if κ1κ2 > 0 then motion is periodic as Gμ = SO(2).
• if κ1κ2 < 0 then motion cen be either bounded or unbounded (depending on μ)
• When is det(μ)=0?
• Is κ2 = -κ1 special in any way?

### 3 vortices

• Is (closure of) J-1(μ) homeo to S3? (and reduced space is S2 and so predict no of REs: 3 collisions (maxima of H) so 2 saddles and a minimum at least)