Isoparametric functions and solutions of Yamabe type equations on manifolds with boundary

TL;DR

This paper investigates the existence and multiplicity of positive solutions to Yamabe-type equations on manifolds with boundary, utilizing isoparametric functions to analyze solutions constant along their level sets.

Contribution

It introduces new existence and multiplicity results for Yamabe equations on manifolds with boundary using isoparametric functions, extending previous work to manifolds with boundary and product manifolds.

Findings

Existence of positive solutions constant along isoparametric level sets.

Multiplicity results for solutions on product manifolds with positive scalar curvature.

Application of isoparametric functions to Yamabe equations on manifolds with boundary.

Abstract

Let $(M,g)$ be a compact Riemannian manifold with non-empty boundary. Provided $f$ an isoparametric function of $(M,g)$ we prove existence results for positive solutions of the Yamabe equation that are constant along the level sets of $f$. If $(M,g)$ has positive constant scalar curvature, minimal boundary and admits an isoparametric function we also prove multiplicity results for positive solutions of the Yamabe equation on $(M \times N,g+th) $ where $(N,h)$ is any closed Riemannian manifold with positive constant scalar curvature.