# Nondegeneracy and the Jacobi fields of rotationally symmetric solutions to the Cahn-Hillard equation

## TL;DR

This paper investigates rotationally symmetric solutions to the Cahn-Hilliard equation in three dimensions, demonstrating their nondegeneracy and analyzing their stability properties through Jacobi fields related to geometric invariances.

## Contribution

The authors establish the nondegeneracy of these solutions and identify the exact number of Jacobi fields, linking stability to the properties of Delaunay surfaces.

## Key findings

- Solutions are nondegenerate with exactly 6 Jacobi fields.
- Jacobi fields correspond to natural invariances and Delaunay parameter variation.
- Stability properties are inherited from Delaunay surfaces.

## Abstract

In this paper we study rotationally symmetric solutions of the Cahn-Hilliard equation in $\mathbb R^3$ constructed by the authors. These solutions form a one parameter family analog to the family of Delaunay surfaces and in fact the zero level sets of their blowdowns approach these surfaces. Presently we go a step further and show that their stability properties are inherited from the stability properties of the Delaunay surfaces. Our main result states that the rotationally symmetric solutions are non degenerate and that they have exactly $6$ Jacobi fields of temperate growth coming from the natural invariances of the problem (3 translations and 2 rotations) and the variation of the Delaunay parameter.