Non-isoparametric solutions of the eikonal equation

TL;DR

This paper classifies quartic polynomial solutions to a specific eikonal equation, showing they are either isoparametric or have a particular explicit form, expanding understanding of solutions beyond classical cases.

Contribution

It provides a complete classification of quartic solutions to the eikonal equation, identifying new explicit solutions not conforming to isoparametric forms.

Findings

Quartic solutions are either isoparametric or of a specific explicit form.

Explicit non-isoparametric solutions are characterized by a particular polynomial structure.

The classification extends the understanding of solution types for the eikonal equation.

Abstract

In this paper, we prove that a quartic polynomial solution of the eikonal equation $|\nabla_x f|^2=16x^{6}$ in $\R{n}$ is either an isoparametric polynomial or congruent to a polynomial $f=(\sum_{i=1}^n x_i^2)^2-8(\sum_{i=1}^k x_i^2)(\sum_{i=k+1}^{n} x_i^2)$, $k=0,1,\...,[\frac{n}{2}]$.