ScalingStacks

Choose $\delta\leq\min(C_{1}^{2},1/2\int_{X}\omega_{0}^{n})$, and for any $t<1$ pick any $p_{t}\in U_{t,\delta}$. If we denote the metric ball of $\omega_{t}$ centered at $p$ and with radius $r$ by $B_{t}(p,r)$, then we get that $U_{t,\delta}\subset B_{t}(p_{t},C_{2})$, where $C_{2}=C_{1}\delta^{-1/2}\geq 1$. Then

$$\int_{B_{t}(p_{t},C_{2})}\omega_{0}^{n}\geq\int_{U_{t,\delta}}\omega_{0}^{n}\geq\frac{1}{2}\int_{X}\omega_{0}^{n},$$

and so

(3.2)

$$\int_{B_{t}(p_{t},C_{2})}\omega_{t}^{n}\geq C_{3}>0,$$

for some constant $C_{3}$ independent of $t$. Since $\mathrm{Ric}(\omega_{t})=0$, the Bishop volume comparison Theorem and (3.2) give that

(3.3)

$$\int_{B_{t}(p_{t},1)}\omega_{t}^{n}\geq\frac{\int_{B_{t}(p_{t},C_{2})}\omega_{t}^{n}}{C_{2}^{2n}}\geq C_{4}>0.$$

The following lemma is due to Yau (see e.g. Theorem I.4.1 in [SY]), but we provide a proof for completeness.