Low energy nodal solutions to the Yamabe equation

TL;DR

This paper constructs low-energy nodal solutions to the Yamabe equation on spheres using isoparametric functions, resulting in solutions with a prescribed number of nodal domains and explicit numerical computability.

Contribution

It introduces a method to generate solutions with any number of nodal domains via a singular ODE reduction and double shooting technique.

Findings

Existence of solutions with exactly k zeros for any positive integer k.

Solutions correspond to isoparametric hypersurfaces as nodal sets.

Low-energy solutions are explicitly computable and have applications in geometric analysis.

Abstract

Given an isoparametric function $f$ on the $n$-dimensional sphere, we consider the space of functions $w\circ f$ to reduce the Yamabe equation on the round sphere into a singular ODE on $w$ in the interval $[0,\pi]$, of the form $w" + (h(r)/\sin r)w'+\lambda(\vert w\vert^{4/n-2}w - w)=0$, where $h$ is a monotone function with exactly one zero on $[0,\pi]$ and $\lambda>0$ is a constant. The natural boundary conditions in order to obtain smooth solutions are $w'(0)=0$ and $w'(\pi )=0$. We show that for any positive integer $k$ there exists a solution with exactly $k$-zeroes yielding solutions to the Yamabe equation with exactly $k$ connected isoparametric hypersurfaces as nodal set. The idea of the proof is to consider the initial value problems on both singularities $0$ and $\pi$, and then to solve the corresponding double shooting problem, matching the values of $w$ and $w'$ at the unique zero of $h$. In particular we obtain solutions with exactly one zero, providing solutions of the Yamabe equation with low energy, which can be computed easily by numerical methods.