Lemma 5.1 [0323]

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Lemma 5.1

Let $X$ , $\omega_{\epsilon}$ be as in Theorem 4.5. Assume $Vol(X)=1$ . Then there exists a function $I(\epsilon)$ depending only on $\epsilon$ and $J$ with $I(\epsilon)\geq C\epsilon^{5}$ , $C$ depending only on $J$ , such that

(1) For any function $f$ on $X$ such that $\int_{X}f\omega_{\epsilon}^{2}=0$ ,

\|df\|^{2}_{2}\geq I(\epsilon)\|f\|^{2}_{4}.

(2) For any function $f$ on $X$ ,

\|df\|^{2}_{2}\geq I(\epsilon)(\|f\|^{2}_{4}-\|f\|^{2}_{2}).