IUMJ

Title: Singular sets of harmonic functions in $\mathbb{R}^2$ and their complexifications in $\mathbb{C}^2$

Authors: Qing Han

Issue: Volume 53 (2004), Issue 5, 1365-1380

Abstract:

It is well known that the critical set of (real) harmonic functions in $B_1 \subset \mathbb{R}^2$ is isolated. In fact, the complex critical set of their holomorphic extensions is also isolated in $\mathbb{C}^2$. It is shown that the number of complex critical set in a fixed ball at $0 \in \mathbb{C}^2$ is bounded by a quadratic order of the frequency of harmonic functions in the (real) unit ball.