Proof. [04BJ]

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Proof.

We will only sketch the modifications. The positivity conditition enters through the monotonicity claim 3.22. Once we drop this, we would allow holomophic discs $u:\Sigma\to X$ that sweep out parts of $L\cup L^{\prime}$ with the reversed orientation. For such curves, claim 3.22 is modified to

Claim 3.29.

Clockwise along $\partial\Sigma$ , the function $\text{Re }F$ is decreasing on the $L$ boundary portion, but increasing on the $L^{\prime}=L_{1}\cup L_{2}$ boundary portion. In particular,

0=\text{Re }F(q)\geq\text{Re }F\geq\text{Re }F(p).

More intrinsically, the real part of the complex volume forms on the moduli spaces at such $u:\Sigma\to X$ are nonpositive.

The corresponding claim 3.23 is modified to

Claim 3.30.

The image $F(\Sigma)\subset\mathbb{C}$ lies below its $L^{\prime}$ boundary portion, and above its $L$ boundary portion.

At almost every point on $L\cup L^{\prime}$ , only automatically transverse holomophic curves pass through it, since by assumption the boundary evaluation of the other holomorphic curves is contained in some subset of $L\cup L^{\prime}$ with Hausdorff dimension $\leq n-1$ . According to the $\pm$ types of the automatically transverse curves, we decompose the weighted characteristic functions $\chi_{A_{i}}$ on $L$ into its positive and negative parts $\chi_{A_{i}^{+}}\geq 0$ and $\chi_{A_{i}^{-}}\leq 0$ .

Then upon integration over the moduli space,

\begin{split}&\text{Re}\int_{L}\chi_{A_{i}^{+}}\Omega=\text{Re}\int_{\mathcal{M}}\tilde{\Omega}_{L_{i}}^{+}\geq 0,\quad\text{Re}\int_{L}\chi_{A_{i}^{-}}\Omega=\text{Re}\int_{\mathcal{M}}\tilde{\Omega}_{L_{i}}^{-}\leq 0,\\ &\text{Im}\int_{L}\chi_{A_{2}^{+}}\Omega\leq\text{Im}\int_{\mathcal{M}}\tilde{\Omega}_{L_{2}}^{+},\quad\text{Im}\int_{L}\chi_{A_{1}^{+}}\Omega\geq\text{Im}\int_{\mathcal{M}}\tilde{\Omega}_{L_{1}}^{+},\\ &\text{Im}\int_{L}\chi_{A_{2}^{-}}\Omega\geq\text{Im}\int_{\mathcal{M}}\tilde{\Omega}_{L_{2}}^{-},\quad\text{Im}\int_{L}\chi_{A_{1}^{-}}\Omega\leq\text{Im}\int_{\mathcal{M}}\tilde{\Omega}_{L_{1}}^{-}.\end{split}

In particular

\arg\int_{L}\chi_{A_{2}^{+}}\Omega\leq\hat{\theta}_{2}^{+},\quad\arg(-\int_{L}\chi_{A_{2}^{-}}\Omega)\leq\hat{\theta}_{2}^{-},

and

\arg\int_{L}\chi_{A_{1}^{+}}\Omega\geq\hat{\theta}_{1}^{+},\quad\arg(-\int_{L}\chi_{A_{1}^{-}}\Omega)\geq\hat{\theta}_{1}^{-}.

Now $\chi_{A_{i}^{+}}\geq 0$ and $\chi_{A_{i}^{-}}\leq 0$ , and the special case where $\chi_{A_{i}^{-}}=0$ almost everywhere is already covered by the positivity condition. The Theorem follows. ∎