ScalingStacks

Corollary 2.3. (Stability estimate) Let $(Y,\omega)$ be a compact Kähler manifold, and $\phi,\psi\in PSH(Y,\omega)\cap C^{0}$, such that $\omega_{\psi}^{n}$ is absolutely continuous. Assume $\left\lVert\phi\right\rVert_{C^{0}}\leq A^{\prime}$ and the Skoda type estimate (3). Then there is a number $B(n,A,A^{\prime},\alpha)$, such that if $\frac{\int_{\psi-\phi\leq-t_{0}}\omega_{\psi}^{n}}{\text{Vol}(Y)}<(2B)^{-2n}$ for some $t_{0}$, then

$$\min(\psi-\phi)\geq-t_{0}-4B\left(\frac{\int_{\psi-\phi\leq-t_{0}}\omega_{\psi}^{n}}{\text{Vol}(Y)}\right)^{1/2n}.$$

.