The Boundary Yamabe Problem, I: Minimal Boundary Case

We apply iteration schemes and perturbation methods to provide a complete solution of the boundary Yamabe problem with minimal boundary scenario, or equivalently, the existence of a real, positive, smooth solution of $-\frac{4(n -1)}{n - 2} \Delta_{g} u + S_{g} u = \lambda u^{\frac{n+2}{n - 2}}$ in $M$, $\frac{\partial u}{\partial \nu} + \frac{n-2}{2} h_{g} u = 0$ on $\partial M$. Thus $g$ is conformal to to the metric $\tilde{g} = u^{\frac{4}{n -2}} g$ of constant scalar curvature $\lambda$ with minimal boundary. In contrast to the classical method of calculus of variations with assumptions on Weyl tensors and classification of types of points on $\partial M$, the boundary Yamabe problem is fully solved here in three cases classified by the sign of the first eigenvalue $\eta_{1}$ of the conformal Laplacian with Robin condition. When $\eta_{1} < 0$, a pair of global sub-solution and super-solution are constructed. When $\eta_{1} > 0$, a perturbed boundary Yamabe equation $-\frac{4(n -1)}{n - 2} \Delta_{g} u_{\beta} + \left( S_{g} + \beta \right) u_{\beta} = \lambda_{\beta} u_{\beta}^{\frac{n+2}{n - 2}}$ in $M$, $\frac{\partial u_{\beta}}{\partial \nu} + \frac{n-2}{2} h_{g} u_{\beta} = 0$ on $\partial M$ is solved with $\beta < 0$. The boundary Yamabe equation is then solved by taking $\beta \rightarrow 0^{-}$. The signs of scalar curvature $S_{g}$ and mean curvature $h_{g}$ play important roles in this existence result.