The Boundary Yamabe Problem, I: Minimal Boundary Case

TL;DR

This paper fully solves the boundary Yamabe problem with minimal boundary in three cases by applying iteration and perturbation methods, providing new solutions without relying on classical variational assumptions.

Contribution

It introduces a comprehensive solution approach for the boundary Yamabe problem in minimal boundary cases, classified by the first eigenvalue of the conformal Laplacian, using iteration and perturbation techniques.

Findings

Solved boundary Yamabe problem for three eigenvalue cases

Constructed global sub- and super-solutions for negative eigenvalues

Solved perturbed boundary Yamabe equations with negative perturbation parameter

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.