Isoparametric Functions and Solutions of Yamabe Type Equations on Manifolds with Boundary

Abstract

Let (M, g) be a compact Riemannian manifold with non-empty boundary. Provided that f is 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 x N, g + th) where (N, h) is any closed Riemannian manifold with positive constant scalar curvature.