Compact balanced threefolds with nonpositive constant Chern holomorphic sectional curvature are Chern flat (c=0) or Kähler (c<0), and constant-curvature LCK manifolds are Kähler or Hopf-covered.
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Compact balanced threefolds and LCK manifolds with constant holomorphic sectional curvature
Compact balanced threefolds with nonpositive constant Chern holomorphic sectional curvature are Chern flat (c=0) or Kähler (c<0), and constant-curvature LCK manifolds are Kähler or Hopf-covered.