7.5 Asymptotic velocity, polarized T3-case

It is of interest to note that the quantity v, or, more precisely, its absolute value, has a geometric significance. Viewing Equations (16View Equation) – (17View Equation) as solutions to the wave-map equation, the kinetic energy density 𝒦 is a geometric object; see Section 6.4. Furthermore, due to the asymptotics of Equation (35View Equation), we have
v2(𝜃) = lim 𝒦 (τ,𝜃). τ→∞

Consequently, v2 is a geometric object. The quantity |v| can also be characterized as the rate at which solutions tend to the boundary of hyperbolic space; if x0 is a fixed point in hyperbolic space and the solution is represented by x, then

dH-(x(τ,𝜃),x0) |v(𝜃)| = τli→m∞ τ , (36 )
where dH is the topological metric induced on hyperbolic space by the hyperbolic metric. To conclude, |v(𝜃)| can be characterized geometrically when viewing Equations (16View Equation) – (17View Equation) as wave-map equations. Furthermore, in the polarized setting, its properties can be used to characterize curvature blow up. We shall loosely refer to v as the velocity.
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