00AE Corollary 4.6. Given small numbers 0<λ,κ≪10<\lambda,\kappa\ll 1, then for tt small enough depending on λ,κ\lambda,\kappa, there exist appropriately chosen local potentials ϕCY,J,t\phi_{CY,J,t} on Log𝒳−1(Int(ΔJ))\text{Log}_{\mathcal{X}}^{-1}(\text{Int}(\Delta_{J})), normalized to inf(ϕCY,J,t−ϕ0∘Log𝒳)=0\inf(\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}})=0, satisfying dμt({ϕCY,J,t−ϕ0∘Log𝒳≥κ/4})<λ,d\mu_{t}(\{\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}\geq\kappa/4\})<\lambda, and ‖ϕCY,J,t−ϕ0∘Log𝒳‖L∞≤C\left\lVert\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}\right\rVert_{L^{\infty}}\leq C independent of λ,κ\lambda,\kappa and small tt.
00AF Proof. By construction in Lemma 4.1, the local potential ϕJ,t\phi_{J,t} of ωFS,t\omega_{FS,t} is ϵ\epsilon-close to ϕ0∘Log𝒳\phi_{0}\circ\text{Log}_{\mathcal{X}}, and since ϵ≪κ\epsilon\ll\kappa these two are practically the same. Up to an overall normalisation constant, which is fixed by inf=0\inf=0, we have ϕCY,J,t=ϕCY,t+ϕJ,t,\phi_{CY,J,t}=\phi_{CY,t}+\phi_{J,t}, so the measure bound follows from the previous result. The uniform L∞L^{\infty} bound follows from Prop. 4.3 and Lemma 4.1 without any reference to λ,κ\lambda,\kappa. ∎