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.

Combining (3.5) with Cheeger-Colding’s segment inequality [4, Theorem 2.11] applied to the function $|d\rho_{t}|_{\omega_{t}}$ we obtain

(3.6)

$$\begin{split}D_{t}(\mu_{t}(A_{1})+\mu_{t}(A_{2}))\int_{X_{t}}|d\rho_{t}|_{\omega_{t}}d\mu_{t}&\geqslant C^{-1}\int_{A_{1}\times A_{2}}\left(\int_{\gamma_{x,y}}|d\rho_{t}|_{\omega_{t}}ds\right)d\mu_{x}d\mu_{y}\\ &\geqslant C^{-1}\int_{A_{1}\times A_{2}}(\rho_{t}(y)-\rho_{t}(x))d\mu_{x}d\mu_{y}\\ &\geqslant\frac{C^{-1}}{3}\mu_{t}(A_{1})\mu_{t}(A_{2}),\end{split}$$

where $D_{t}=\mathrm{diam}(X_{t},\omega_{t})$, and in the $\int_{A_{1}\times A_{2}}$ we are actually only integrating over the subset of pairs $(x,y)$ which are joined by a unique $\omega_{t}$-minimal geodesic, which has full measure (cf. [4]).

Combining (3.4) and (3.6) gives

$$\mu_{t}(A_{1})\mu_{t}(A_{2})\leqslant CD_{t}(\mu_{t}(A_{1})+\mu_{t}(A_{2}))\leqslant CD_{t}.$$

Lastly, from the definition of $\rho_{t}$ and from (3.1), a direct computation in polar coordinates (analogous to the one in [3]) gives

$$\mu_{t}(A_{1})\geqslant C^{-1},\quad\mu_{t}(A_{2})\geqslant C^{-1},$$

for a fixed constant $C$, and so $D_{t}\geqslant C^{-1},$ as desired.

∎