ScalingStacks

Proof of the diameter lower bound in Theorem 1.1. Thanks to Proposition 3.1, on $B_{t}\subset X_{t}$ we have

$$|d\rho_{t}|_{\omega_{t}^{\prime}}^{2}\leqslant C,$$

for some constant $C$ independent of $t$. We then use this together with the elementary inequality $|d\rho_{t}|_{\omega_{t}}^{2}\leqslant|d\rho_{t}|_{\omega_{t}^{\prime}}^{2}\Tr_{\omega_{t}}\omega_{t}^{\prime}$ to get

$$\left(\int_{X_{t}}|d\rho_{t}|_{\omega_{t}}d\mu_{t}\right)^{2}=\left(\int_{B_{t}}|d\rho_{t}|_{\omega_{t}}d\mu_{t}\right)^{2}\leqslant\int_{B_{t}}|d\rho_{t}|_{\omega_{t}}^{2}d\mu_{t}\leqslant C\int_{X_{t}}\Tr_{\omega_{t}}\omega_{t}^{\prime}d\mu_{t},$$

while from (2.1) we get

$$\int_{X_{t}}\Tr_{\omega_{t}}\omega_{t}^{\prime}d\mu_{t}=\frac{n\int_{X_{t}}\omega^{\prime}_{t}\wedge\omega_{t}^{n-1}}{\int_{X_{t}}\omega_{t}^{n}}=\frac{n\int_{X_{t}}c_{1}(\mathfrak{L})\cdot c_{1}(L)^{n-1}}{\int_{X_{t}}c_{1}(L)^{n}}\leqslant C,$$

and so

(3.4)

$$\int_{X_{t}}|d\rho_{t}|_{\omega_{t}}d\mu_{t}\leqslant C.$$

Define two subsets of $X_{t}$ by $A_{1}=\{\rho_{t}<-\frac{2}{3}\}$ and $A_{2}=\{-\frac{1}{3}\leqslant\rho_{t}\leqslant 0\}$. Given two points $x\in A_{1},y\in A_{2}$ which are connected by a unique minimal geodesic $\gamma_{x,y}$ (w.r.t. $\omega_{t}$), we can bound

(3.5)

$$\rho_{t}(y)-\rho_{t}(x)\leqslant\int_{\gamma_{x,y}}|d\rho_{t}|_{\omega_{t}}ds,$$

where $\gamma_{x,y}$ is parametrized with respect to $\omega_{t}$-arclength.