The Boundary Yamabe Problem, I: Minimal Boundary Case

Abstract

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.