Nonuniqueness for a fully nonlinear boundary Yamabe-type problem via bifurcation theory

TL;DR

This paper explores the nonuniqueness of solutions to a fully nonlinear boundary Yamabe-type problem on compact manifolds with boundary, using bifurcation theory to construct examples with multiple solutions.

Contribution

It introduces a bifurcation theorem for a nonlinear elliptic boundary value problem and constructs examples of manifolds with multiple solutions, extending understanding of boundary Yamabe problems.

Findings

Multiple non-homothetic solutions exist for certain boundary Yamabe problems.

Bifurcation theory can be applied to fully nonlinear elliptic boundary value problems.

Examples include manifolds with boundary as a product of a sphere and an Einstein manifold.

Abstract

One way to generalize the boundary Yamabe problem posed by Escobar is to ask if a given metric on a compact manifold with boundary can be conformally deformed to have vanishing $\sigma_k$-curvature in the interior and constant $H_k$-curvature on the boundary. When restricting to the closure of the positive $k$-cone, this is a fully nonlinear degenerate elliptic boundary value problem with fully nonlinear Robin-type boundary condition. We prove a general bifurcation theorem which allows us to construct examples of compact Riemannian manifolds $(X,g)$ for which this problem admits multiple non-homothetic solutions in the case when $2k<\dim X$. Our examples are all such that the boundary with its induced metric is a Riemannian product of a round sphere with an Einstein manifold.