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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 ofcomponents of n-dimensional p-adic potentially semistable local Galoisdeformation rings that are seen by potentially automorphic compatible systemsof polarizable Galois representations over some CM field is independent of theparticular global situation. We also (under the same assumption on n) improveon the main potential automorphy result of [BLGGT14b], replacing "potentiallydiagonalizable" by "potentially globally realizable".

Globally realizable components of local deformation rings