### 7.5 Asymptotic velocity, polarized T^{3}-case

It is of interest to note that the quantity , or, more precisely, its absolute value, has a geometric
significance. Viewing Equations (16) – (17) as solutions to the wave-map equation, the kinetic energy
density is a geometric object; see Section 6.4. Furthermore, due to the asymptotics of Equation (35),
we have
Consequently, is a geometric object. The quantity can also be characterized as the rate at which
solutions tend to the boundary of hyperbolic space; if is a fixed point in hyperbolic space and the
solution is represented by , then

where is the topological metric induced on hyperbolic space by the hyperbolic metric. To conclude,
can be characterized geometrically when viewing Equations (16) – (17) as wave-map equations.
Furthermore, in the polarized setting, its properties can be used to characterize curvature blow up. We shall
loosely refer to as the velocity.