Gradient map of isoparametric polynomial and its application to Ginzburg-Landau system

TL;DR

This paper investigates the properties of the gradient map of isoparametric polynomials, revealing its mappings between hypersurfaces and focal submanifolds, and applies these findings to the Ginzburg-Landau system, providing new insights and counterexamples.

Contribution

It explicitly calculates the gradient map of isoparametric polynomials and explores its geometric properties and applications to the Ginzburg-Landau system, including a counterexample to a symmetry question.

Findings

Gradient map maps isoparametric hypersurfaces to focal submanifolds.

Gradient map is a homogeneous polynomial automorphism on certain hypersurfaces.

Provides a counterexample to the symmetry question in the Ginzburg-Landau system in dimension 6.

Abstract

In this note, we study properties of the gradient map of the isoparametric polynomial. For a given isoparametric hypersurface in sphere, we calculate explicitly the gradient map of its isoparametric polynomial which turns out many interesting phenomenons and applications. We find that it should map not only the focal submanifolds to focal submanifolds, isoparametric hypersurfaces to isoparametric hypersurfaces, but also map isoparametric hypersurfaces to focal submanifolds. In particular, it turns out to be a homogeneous polynomial automorphism on certain isoparametric hypersurface. As an immediate consequence, we get the Brouwer degree of the gradient map which was firstly obtained by Peng and Tang with moving frame method. Following Farina's construction, another immediate consequence is a counter example of the Br\'ezis question about the symmetry for the Ginzburg-Landau system in dimension 6, which gives a partial answer toward the Open problem 2 raised by Farina.