QUASI-FUCHSIAN FLOWS AND THE COUPLED VORTEX EQUATIONS

The paper establishes Theorem 1.1 by (i) defining G([σ],A)=(Φ−1[σ](A),Φ−1[σ](-A)), showing Diff0(M)-equivariance via Φ’s naturality, and proving bijectivity by minimizing the sum of two Dirichlet energies and invoking uniqueness of the critical point (Proposition 3.1) ; (ii) constructing an explicit principal-bundle isomorphism I:SMg→SMh and deriving the conjugacy identities I*α1=(1+½Vλ)α−λβ, I*β1=−λα+(1−½Vλ)β, I*ψ1=ψ to match weak foliations for rs,ru=−1+½Vλ, 1+½Vλ (Items a–c) ; (iii) invoking [MP19, Thm. 5.5] to get volume preservation iff [g1]=[g2] iff A=0 (Item d) ; and (iv) proving the length-spectrum identities ℓg1(c)=ℓg(π∘ζ)+½∫ζVλ and ℓg2(c)=ℓg(π∘ζ)−½∫ζVλ, yielding ℓg(π∘ζ)=½(ℓg1(c)+ℓg2(c)) and uniqueness of a closed orbit in each free homotopy class (Item e) . The candidate solution proves the same result but via a different route: it uses the minimal Lagrangian/holomorphic data to produce (g,A) and explicit coframes/foliation formulas, computes divΩF=Vλ to characterize volume preservation, and derives the length identity from rs,ru and ru−rs≡2. These steps align with standard facts recalled in the paper (e.g., rs,ru in (1.4) and Vλ as divergence) and yield the same conclusions. Thus both are correct; the proofs are methodologically different (PDE/energy minimization plus explicit bundle isomorphism I in the paper vs. minimal Lagrangian/coframe–Riccati machinery in the model).