Various sources touch briefly on the reverse mathematics of measure theory and complex analysis. But I have found none on the uniformization theorem for Riemann surfaces or the existence of non-constant meromorphic functions. The proofs of those results use very penetrating analysis but I do not know if they need high logical strength. For references on the usual proofs see 
[The existence of meromorphic functions on Riemann surfaces](https://mathoverflow.net/questions/125659/the-existence-of-meromorphic-functions-on-riemann-surfaces).


Is the reverse math of these theorems known?



Reverse mathematics of meromorphic functions on Riemann surfaces
I do not know about the full uniformization theorem, but reverse mathematics of Riemann mapping theorem (by applying non-standard second-order arithmetics) is done in MR3129726 
Horihata, Yoshihiro; Yokoyama, Keita(J-AIST-SIF)
Nonstandard second-order arithmetic and Riemann's mapping theorem. (English summary)
Ann. Pure Appl. Logic 165 (2014), no. 2, 520–551. References of this paper may also be useful.


Regarding uniformization theorem from the viewpoint of computability theory, see MR2983724 Rettinger, Robert(D-HGNMC) Compactness and the effectivity of uniformization. (English summary) How the world computes, 626–625, Lecture Notes in Comput. Sci., 7318, Springer, Heidelberg, 2012.


