Bifurcation results for the Yamabe problem on Riemannian manifolds with boundary

TL;DR

This paper uses bifurcation theory to demonstrate the existence of infinitely many conformal classes with multiple non-homothetic metrics of null scalar curvature and constant boundary mean curvature on certain product manifolds.

Contribution

It introduces new bifurcation results for the Yamabe problem on product manifolds with boundary, establishing the existence of multiple conformal metrics with specified curvature properties.

Findings

Existence of infinitely many conformal classes with desired curvature properties.

Multiple non-homothetic solutions within these classes.

Application of bifurcation theory to boundary Yamabe problems.

Abstract

We consider the product of a compact Riemannian manifold without boundary and null scalar curvature with a compact Riemannian manifold with boundary, null scalar curvature and constant mean curvature on the boundary. We use bifurcation theory to prove the existence of a infinite number of conformal classes with at least two non-homothetic Riemannian metrics of null scalar curvature and constant mean curvature of the boundary on the product manifold.