On closed surfaces there are three basic ways to evolve a metric, by conformal change, by pull-back with diffeomorphisms and by horizontal curves, moving orthogonally to the first two types of evolution. As we will discuss in this talk, horizontal curves are very well behaved even if the underlying conformal structures degenerate in moduli space as t to T. We can describe where the metrics will have essentially settled down to the limit by time t T as opposed to regions on which the metric still has to do an infinite amount of stretching. This quantified information is essential in applications and allows us to prove a "no-loss-of-topology" result at finite time singularities of Teichmüller harmonic map flow which, combined with earlier work, yields that this geometric flow decomposes every map into a collection of branched minimal immersions and curves. This is joint work with Peter Topping