The Yamabe Problem and Nonlinear Boundary Value Problems

We study the Yamabe problem in the context of manifolds with boundary - a basic problem in Riemannian geometry - from the point of view of nonlinear elliptic boundary value problems. By making good use of bifurcation theory from a simple eigenvalue, we show that nonpositive scalar curvatures and nonpositive mean curvatures are not always conformal to constant negative scalar curvatures and the zero mean curvature.