As I understand it Uber's profits from the US market (and VC money) is funding the global expansion. If Uber lose their dominant position in the US we should expect their plans in the rest of the world to stop quite quickly; they won't be able to afford them.
Yeah, this is pretty key. If Uber's dominant position in the US is slipping but there's still a clear way for them to emerge on top globally and cash is the limiting concern, it seems likely they'll be able to continue subsidizing international expansion with more VC money.