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.
Steinke, and Koen Struyve, On metrically complete B ruhat- T its buildings , Adv
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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.