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

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 Hk-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