For reductive group scheme actions on projective schemes over complete non-archimedean fields, the minimal invariant locus coincides with the semistable reduction translation locus and is non-empty under mild completeness assumptions.
I [Springer, Berlin, 1993; MR1261420 (95m:90001)] and II [ibid.; MR1295240 (95m:90002)]
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Semistable reductions and minimalities of invariants for group scheme actions on projective schemes
For reductive group scheme actions on projective schemes over complete non-archimedean fields, the minimal invariant locus coincides with the semistable reduction translation locus and is non-empty under mild completeness assumptions.