  Let n be either 2, or an odd integer greater than 1, and fix a prime p > 2(n
+ 1). Under standard "adequate image" assumptions, we show that the set of
components of n-dimensional p-adic potentially semistable local Galois
deformation rings that are seen by potentially automorphic compatible systems
of polarizable Galois representations over some CM field is independent of the
particular global situation. We also (under the same assumption on n) improve
on the main potential automorphy result of [BLGGT14b], replacing "potentially
diagonalizable" by "potentially globally realizable".
