Sorry. I was being silly.

Yes x is the norm of the Teichmuller map. Sorry, I am a novice in this field. So you are saying that if the $L^{\infty}$ norms of some Beltramis get close to the extremal $L^{\infty}$, then the corresponding q.c maps get close in the uniform topology (also I don't want just a subsequence but the entire sequence to converge)? If so, can you cite a reference. Thanks a million!

For bounded domains in Euclidean space, this is true (Evans' book). It gets tricky when one wants to approximate by smooth functions that smooth upto the boundary.

This isn't a complete answer, but may help. For smooth u_k, u_k >v is an open set. For non-smooth u, this is not necessarily the case. It may have a boundary. Just outside the boundary max(u,v) = v and inside max(u,v) = u. So, whilst testing the current against test functions whose support "ends" at the boundary, it isn't obvious (to me) that dd^c(max(u,v)) = dd^c (u).