  The Teichm\"uller harmonic map flow deforms both a map from an oriented
closed surface $M$ into an arbitrary closed Riemannian manifold, and a constant
curvature metric on $M$, so as to reduce the energy of the map as quickly as
possible [16]. The flow then tries to converge to a branched minimal immersion
when it can [16,18]. The only thing that can stop the flow is a finite-time
degeneration of the metric on $M$ where one or more collars are pinched. In
this paper we show that finite-time degeneration cannot happen in the case that
the target has nonpositive sectional curvature, and indeed more generally in
the case that the target supports no bubbles. In particular, when combined with
[16,18,9], this shows that the flow will decompose an arbitrary such map into a
collection of branched minimal immersions.
