Theorem 6 Consider a solution to Equations (11)–(12). Let and let be the metric induced by the Riemannian metric given by Equation (22). Then there is a , a and a curve such that

for all . The possibilities for are as follows.

- If all the constants and are zero, is a point.
- If , but the constants are not all zero, is either a horocycle (i.e., a circle touching the boundary) or a curve .
- If , is either a circle intersecting the boundary transversally or a straight line intersecting the boundary transversally.
- If , is a circle inside the upper half plane.

Furthermore, it is possible to describe in detail the behavior of the solutions along the circles; see [75, Theorem 1.3–1.5, pp. 662–664] as well as [75, Figure 1.1–1.3, pp. 663–664]. In fact, the solution tends to the boundary along the circle when and , and are not all equal to zero. In the case of , the solution oscillates around the circle forever and is asymptotically periodic with respect to a logarithmic time coordinate.

http://www.livingreviews.org/lrr-2010-2 |
This work is licensed under a Creative Commons License. Problems/comments to |