## Category: Blog

- The Calabi-Yau equation (2) ()
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 […]

- The Calabi-Yau equation (1) ()
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 […]

- Balanced manifolds and the space of volume forms ()
This post aims to give an overview of a paper I recently uploaded to arxiv. M. L. Michelsohn in the 80s introduced the notion of a balanced metric on a complex manifold. The definition can be simply written as a Hermitian metric $\omega$ that satisfies $d \omega^{n-1} = 0 $. These metrics are more abundant […]