6.3.3 Real MA metric [004Z]

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6.3.3 Real MA metric

To complete the circle, we need

Lemma 6.9.

The limiting convex potentials $\bar{\phi}_{J,0}$ solve the real MA equation (17) on the interior of $\Delta_{J}$ .

The strategy is to pass the complex MA equation on $U_{J,t}$ to the limit. This is feasible, morally because the complex MA operator is weakly continuous under the $C^{0}$ -convergence of potentials. In our setting, an extra subtlety is that the sequence of potentials are defined on different manifolds, and the modification of the usual arguments are carried out in [52, section 5.1].