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} Δ_{g} u + S_{g} u = λu^{\frac{n+2}{n - 2}} $ in $ M $, $ \frac{\partial u}{\partial ν} + \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 $ λ$ 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 $ η_{1} $ of the conformal Laplacian with Robin condition. When $ η_{1} < 0 $, a pair of global sub-solution and super-solution are constructed. When $ η_{1} > 0 $, a perturbed boundary Yamabe equation $ -\frac{4(n -1)}{n - 2} Δ_{g} u_β + \left( S_{g} + β\right) u_β = λ_β u_β^{\frac{n+2}{n - 2}} $ in $ M $, $ \frac{\partial u_β}{\partial ν} + \frac{n-2}{2} h_{g} u_β = 0 $ on $ \partial M $ is solved with $ β< 0 $. The boundary Yamabe equation is then solved by taking $ β\rightarrow 0^{-} $. The signs of scalar curvature $ S_{g} $ and mean curvature $ h_{g} $ play important roles in this existence result.