UNIQUE ERGODICITY FOR SINGULAR HOLOMORPHIC FOLIATIONS OF P3(C) WITH AN INVARIANT PLANE

The paper’s Theorem 4 states and proves that for a singular holomorphic foliation F of P3(C) with hyperbolic singularities, an invariant plane P, and no non-zero foliation cycles directed by F|P, there is a unique mass-1 harmonic current directed by F, namely the zero-extension to P3(C) of the unique harmonic current on P2(C) directed by F|P (explicitly stated in Theorem 4 and concluded after Theorem 7) . The model gives a shorter analytic proof: (i) use Fornæss–Sibony’s uniqueness on P2(C) (quoted as Theorem 2 in the paper) to get the unique current on P, (ii) extend by zero to P3(C), and (iii) show any directed harmonic current on P3(C) has no mass in P3\P by testing against dd^c of a convex exhaustion χ∘φ on C3. With a small fix (compare positive leafwise forms directly to dd^c(χ∘φ) on compact sublevel sets rather than to dd^cφ itself), the model’s argument is sound and reaches the same conclusion. The two approaches are substantively different: the paper uses Brownian motion and Lyapunov/contraction/similarity machinery, whereas the model uses a plurisubharmonic-exhaustion-and-positivity argument. Hence: both correct, different proofs.