Explorations

In this post I will explain the second part of my paper titled “Volume forms on balanced manifolds and the Calabi-Yau equation”.

First, a bit of history of this theory. Arguably, the most important paper in the field of geometric PDEs on complex manifolds came from Yau’s breakthrough in 1976. In the 50’s Calabi conjectured the existence of special Kähler metrics whose Ricci curvature vanishes everywhere.

To motivate such a problem, take the simple example of a constant curvature surface in $\mathbb R^n$. This could be sphere, plane… On a curved manifold, these shapes are much more difficult to figure out, and usually involves solving a highly nonlinear PDE.

Calabi showed that this problem can be reduced to solving the following PDE

$$\det (\omega + \sqrt{-1} \partial \bar{\partial} u) = e^F det(\omega),$$

called the complex Monge-Ampere equation, for a background Kähler metric $\omega$. Here $u$ is the unknown function. Calabi worked on this problem and was able to show uniqueness of the solution if it exists. But the hard estimates required for showing existence was not obtained (although some of Calabi’s original computations were used in the proof that later appeared). Eugenio Calabi sadly passed away this year. I got the opportunity to attend his memorial seminar, where many contributions of Calabi were explained. It is clear that Calabi was the driving force behind many of the most active research topics in this area. I will explore some of them in detail later.

S.T. Yau finally completed Calabi’s program by deriving all the estimates required to show existence. We will not go through Yau’s methods now, but just point out that there are much simpler ways of getting $C^0$, and $C^1$ estimates now. For $C^3$ estimates, Evans-Krylov theorem is used nowadays in place of the long computations of Yau. But the $C^2$ estimates cannot be simplified too much.

In the study of complex manifolds, a Kähler metric is not always the most useful notion (although it is the nicest and closest to Riemannian metrics), as many manifolds do not admit them. From the 80s, mathematicians were searching for other metrics which are less restrictive. One such metric was discovered by Paul Gauduchon, which is named after him. A Gauduchon metric satisfies $\partial \bar{\partial} \omega^{n-1} = 0$, in place of $d \omega =0$ for a Kähler metric. It can be shown that all complex manifolds admit Gauduchon metrics. This by itself is a very useful fact, since we could get many of the similar computations using Gauduchon metric as using Kähler metrics.

Gauduchon asked whether a Calabi-Yau type theorem can be proved for Gauduchon metrics. That is, Can you find a Gauduchon metric $\tilde{\omega}$, whose curvature form vanishes.

$$- \sqrt{-1} \partial \bar{\partial} \log \det{\tilde{\omega}} = 0$$

Similar to Calabi, we could perform some linear algebra and calculus tricks to transform this into the following nonlinear PDE.

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

where $X$ is a function involving the torsion tensor of $\omega$ and gradient of $u$. It is also linear in $\nabla u$. $X$ can be explicitly written down and its form is important in solving this equation.

This post will be continued later.