# Harmonic cubic homogeneous polynomials such that the norm-squared of the Hessian is a multiple of the Euclidean quadratic form

## TL;DR

This paper classifies harmonic cubic homogeneous polynomials with Hessian norm-squared proportional to the Euclidean form, providing solutions in all dimensions and a complete classification in dimensions up to four.

## Contribution

It offers a comprehensive classification of such polynomials and methods to distinguish inequivalent solutions under conformal transformations.

## Key findings

- Solutions constructed in all dimensions
- Complete classification in dimensions up to 4
- Techniques for inequivalence detection

## Abstract

There is considered the problem of describing up to linear conformal equivalence those harmonic cubic homogeneous polynomials for which the squared-norm of the Hessian is a nonzero multiple of the quadratic form defining the Euclidean metric. Solutions are constructed in all dimensions and solutions are classified in dimension at most $4$. Techniques are given for determining when two solutions are linearly conformally inequivalent.