§ 5.41 Isometric curve-shortening solitons in the hyperbolic plane6E7783E99D53

For every nonzero Minkowski vector $\widetilde v\in \mathbb R^{2,1}$, Woolgar–Xie give a two-parameter family of nontrivial curve-shortening solitons in $\mathbb H^2$. Each is complete, unbounded and properly embedded, with parameter domain $\mathbb R$ and two ends. Its motion is the one-parameter group of hyperbolic isometries determined by the corresponding Killing generator; timelike, null and spacelike types correspond to elliptic, parabolic and hyperbolic motions. The ends asymptote to horocycles or geodesics under the restrictions in Theorem 1.2. In particular both ends cannot be geodesic-asymptotic, and the timelike case cannot have a geodesic-asymptotic end. The curvature equation is $k_ g=\langle K,N\rangle $; these are eternal isometric flows, not Euclidean shrinkers or homothetically scaled hypercycles. The figure uses a fixed hyperbolic disk or half-plane chart and compares the group motion at three times. Its profile is schematic; no Euclidean curvature conclusion follows from its appearance in the conformal model. Euclidean entropy is inapplicable.

Sources: [WoolgarXie2022].