ScalingStacks

Here we present a second strategy for the potential estimate, which aims to circumvent the uniform $L^{1}$-stability estimate Theorem 2.6, and we explain why there is a difficulty with this second approach. Readers who wish to follow the main line of the proof may skip this section.

An outline of this strategy is

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  Choose some suitable integral normalisation on $\phi_{CY,t}$. Apply Poincaré inequality to prove the average $L^{2}$-integral of $\psi_{t}-\phi_{CY,t}$ is small in the generic region $W_{\delta}$, which occupies most of the $d\mu_{t}$-measure.

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  Deduce the measure is small on the set where $\phi_{CY,t}-\psi_{t}$ is perceptibly negative.

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  Apply the stability estimate Cor. 2.3 to show $\phi_{CY,t}-\psi_{t}$ cannot be perceptibly negative. Since $\left\lVert\psi_{t}\right\rVert_{C^{0}}$ is negligible, it shows the minimum of $\phi_{CY,t}$ is almost zero.

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  Using the small $L^{2}$-average bound on $\phi_{CY,t}$, one applies the mean value inequality to derive a small upper bound on $\phi_{CY,t}$ in the generic region $W_{\delta}$.

The problem lies in the fact that the $n$-dimensional open faces $\text{Int}(\Delta_{J})$ of $Sk(X)$ are disconnected, so the average values on each face are a priori unrelated. Thus the Poincaré inequality can only imply a small bound on $\phi_{CY,t}-\psi_{t}$ with a priori different normalisations associated to each open face, which is not good enough to get a small bound on average $L^{2}$-integral of $\psi_{t}-\phi_{CY,t}$.