Abstract: In 2024, we constructed a singular solution to the complex Monge–Ampère equation with constant right-hand side. In this talk, we explain the mechanism underlying that construction and show how it yields new examples with more severe singularities.

For two symmetry classes in complex dimension two, a partial Legendre transform relates the complex Monge–Ampère equation to the Laplace equation. This correspondence allows us to construct plurisubharmonic solutions of $\det(u_{i\bar j})=1$ in the Bedford–Taylor sense from superpositions of harmonic functions.

In every complex dimension $n\geq2$, these include:

1. A bounded solution on the unit polydisk $\mathbb D^n$ that is discontinuous almost everywhere and nowhere classically differentiable.

2. A continuous exhaustion function on $\mathbb C^n$ that is not in $W^{2,1}_{\mathrm{loc}}(\mathbb C^n)$. This example disproves a conjecture of Błocki on regularity in hyperconvex domains and a conjecture of B. Chen on the completeness of metrics induced by proper entire solutions.

3. A solution on $\mathbb D^n$, continuous up to the boundary, whose distributional Laplacian has a nonzero singular part.

We also revisit the Taub–NUT metric, a classical gravitational instanton, and discuss its relation to the quadratic polynomial via the partial Legendre transform.

This is joint work with Wenlong Wang.

Brief biography: Jiaxiang Wang is currently an associate professor at Nankai University. He obtained his Ph.D. from Zhejiang University and was a postdoc at the Beijing International Center for Mathematical Research, Peking University, from 2021 to 2023. His research focuses on complex geometry, complex Monge–Ampère equations, and pluripotential theory.

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