A new series: Complex MA-type equations

Published March 24, 2021 (updated: March 29, 2021) in Balanced metrics, Estimates, Monge-Ampere, PDE.

This is the first post in a new series based on the recent topics I’m reading. The theme will be to study complex Monge-Ampere type equations that will solve important problems in geometry. A typical and famous example of this would be the Calabi-Yau theorem. This was proved by Yau in 1978 and since then inspired many other results of the same nature, whether it be extending to the Kahler-Einstein case or, solving non-Kahler variants of this problem. In this series, we will be mostly interested in the latter. That is, we look at the cases when the manifold may not admit a Kahler metric, but does admit other types of hermitian metrics.

For this post, the main focus would be to solve the exact same Monge-Ampere equation as in the CY theorem, but by only making the following assumptions on the background hermitian metric:

a) For n=2, we make no assumptions on the metric.

b) For n>2, the metric is balanced, $d\omega^{n-1}=0$ .

The general plan in all such problems is to use the continuity method. This technique will be described in detail in a different post. But the crucial element of this method is finding a uniform apriori estimate for the $C^{2,\alpha}$ Holder norm of the solution (for some $0<\alpha\leq 1$ , the exact value of $\alpha$ is not important). We separate this into problems of finding $C^0$ , $C^2$ and $C^{2, \alpha}$ estimates rather than doing all in one step. I have intentionally skipped the $C^1$ estimates here, since that always follow from interpolation inequalities once we have the $C^0$ and $C^2$ estimates. See for example the 6th chapter of [GT].

The problem above was solved by Valentino Tosatti and Ben Weinkove in [TW]. The interesting fact about their solution is that they show a $C^2$ estimate in terms of the $C^0$ norm of the solution: $tr_{g}g'\leq Ae^{\phi-\inf(\phi)}$ . This can then be used to get a $C^0$ estimate without even using the initial PDE again.

The $C^2$ estimates are done first by estimating $tr_gg'$ , where $g,g'$ are the given metric and the new metric respectively ( $g'=g+\sqrt{-1}\partial \bar{\partial }\phi$ ). This is done by applying maximum principle to the function $\log(tr_gg')-A\phi$ as in Yau’s solution. The idea is all the same. But now there are some differences in calculations. For one, there is no holomorphic normal coordinates as in the Kahler case. So all the first derivatives of $g$ appears in the expansion of $\Delta\log(tr_gg')$ . This issue is partly solved by a version of normal co-ordinates for hermitian metrics given by Guan-Li:

At any point $p$ , there exists holomorphic normal coordinates such that $g_{i\bar j}(p)=\delta_{ij}$ and $\partial_{j}g_{i\bar i}(p)=0$ for all $i,j$ and $\partial_i\partial_{\bar j} \phi(p)$ is diagonal.

Note that $\Delta'\log tr_{g}g'=\frac{\Delta'tr_gg'}{tr_gg'}-\frac{|\partial tr_gg'|^2_{g'^2}}{(tr_gg')^2}$ .

The first term of RHS can be estimated the same way for both cases (a), (b). The fourth order derivative of $\phi$ that appears in this calculation is transformed to a product of first order derivatives of $g'$ and some estimated quantities by using the equation (original PDE). This is then used to absorb the troublesome terms that appear due to the non-Kahlerness as described in the previous paragraph.

For the second term of RHS above, the estimation is done differently for each case. For (a), make use of the hermitian normal coordinates to write $\partial_i tr_gg'=\sum_j \partial_i g'_{j\bar j}$ . Then since the dimension is $2$ we can show that $tr_gg'$ and $tr_{g'}g$ are uniformly equivalent (are constant multiples of each other). Now performing several Cauchy-Schwarz arguments we will get the required estimate.

For (b), things will be much more simpler due to the following observation. The balanced condition $d\omega^{n-1}=0$ is equivalent to the vanishing of trace of the torsion tensor $\sum_{i}T_{ij}^i=0$ . This is also same as $\sum_j \partial_j g_{i\bar j}=0$ . Now $tr_gg'=\sum_j \partial_j g'_{i\bar j}$ follows and the required inequality is obtained in one step from here.

The usual maximum principle arguments can be used to get the $C^2$ estimate from an elliptic inequality for $tr_gg'$ . See [TW] for all the details. It is interesting how they find the $C^0$ estimates just from the special form of $C^2$ estimate we have here. In fact, this technique can be applied to any hermitian metric $\omega$ to obtain $C^0$ and higher order estimates if that particular form of $C^2$ estimate can be shown first.

The first step in doing this is to use a Moser iteration argument on the exponential of the function $\phi$ .