Explorations

This is a continuation from a previous post. We rewrite the equation here.

$$\det( \omega_0 + \frac{1}{n-1}(\Delta u \omega – \sqrt{-1} \partial \bar{\partial} u) + X[\nabla u]) = e^F \det{\omega}.$$

The equation from Gauduchon conjecture is also called the Monge-Ampere equation for $(n-1)$ plurisubharmonic functions. This is because it involves the form $(\Delta u \omega – \sqrt{-1} \partial \bar{\partial} u)$ , which in orthonormal coordinates that diagonalizes the Hessian of $u$, is the sum of $(n-1)$ eigenvalues of the Hessian. Also the form inside the parenthesis needs to be positive for this to represent a metric. Hence this can be seen as a type of plurisubharmonicity of $u$.

This equation without the term $X[\nabla u]$ has been solved by Tosatti-Weinkove in 2013. Geometrically, this is the case when the manifold is Kähler and the torsion terms vanish. This paper provides the basic ideas for handling this new type of Monge-Ampere equation, but the Gauduchon conjecture is still not solved. Gauduchon manifolds need not be Kähler, and hence we cannot ignore the torsion terms.

In a later work in 2015, the equation above was completely solved by Szekelyhidi-Tosatti-Weinkove. They used the special form of the term $X[u]$ to deal with the gradient terms in the equation. This is a remarkable achievement from the point of view of nonlinear PDE theory. But still it remains to understand the geometric implications of such a theorem.

Yau later discussed the importance of proving such a theorem for balanced metrics in place of Gauduchon metrics. One reason is that the Gauduchon condition $\partial \bar{\partial} \omega^{n-1} = 0$ is a scalar condition (one differential equation), but the balanced metric condition $d \omega = 0$ is a system of equations with a wider range of possibilities. This might be more useful in applications. There are other contexts in which balanced metrics are important. I will give detailed descriptions of this in a later post. For now, we will focus on the PDE problem of solving the Calabi-Yau theorem assuming that the metric is only balanced. In this case, the equation differs in only one term.

$$\det( \omega_0 + \frac{1}{n-1}(\Delta u \omega – \sqrt{-1} \partial \bar{\partial} u) + X[\nabla u] + \star\sqrt{-1} \partial \bar{\partial} \omega^{n-2} u) = e^F \det{\omega}.$$

But this one term can cause significant issues in solving this equation. Let us denote the $(1,1)$ form $E = \star\sqrt{-1} \partial \bar{\partial} \omega^{n-2}$. The difficult part is to derive $C^0$ estimates. All the other estimates will follow similar to the previous papers involving $(n-1)$ Monge-Ampere equations. In my paper, it was showed that this can be done under the assumption that $E \leq 0$ as a $(1,1)$ form. The condition that $\partial \bar{\partial} \omega^{n-2} =0$ is the astheno-Kähler condition of Jost-Yau. Based on this, I called metrics that satisfy this condition to be sub-astheno-Kähler metrics. It would be interesting to know the relevance of this notion and the possible restrictions to it.