ScalingStacks

Theorem 1.4. (Uniform $L^{\infty}$-estimate) Let $\phi_{t}$ be the Kähler potential of the Calabi-Yau metric in the class $(X_{t},[\omega_{t}])$, namely

$$\frac{(\omega_{t}+\sqrt{-1}\partial\bar{\partial}\phi)^{n}}{\int_{X_{t}}\omega_{t}^{n}}=d\mu_{t},\quad\sup_{X_{t}}\phi_{t}=0.$$

Then $\left\lVert\phi_{t}\right\rVert_{L^{\infty}}\leq C$ independent of $t$ for $0<|t|\ll 1$.