The Boundary Yamabe Problem, II: General Mean Curvature Case

We apply iterative schemes and perturbation methods to completely solve the boundary Yamabe problem under the most general 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 = \frac{n-2}{2} \zeta u^{\frac{n}{n - 2}} $ on $ \partial M $ with some $ \lambda $ and $ \zeta $. The boundary Yamabe problem is solved under the classification of the sign of the first eigenvalue $ \eta_{1} $ of the conformal Laplacian with homogeneous Robin condition. The signs of scalar curvature $ S_{g} $ and mean curvature $ h_{g} $ play an important role in this existence result. In contrast to the classical method, Weyl tensor and classification of boundary points play no role in this article.Comment: 23 pages. Results in arXiv:2011.15436 (Yamabe Problem on Closed Manifolds), especially Proposition 3.3 and Theorem 4.3 there, play a central role in this article. See latest version of arXiv:2011.15436 (Jan. 10th version, in which issues in Theorem 4.3, Theorem 4.4 and Appendix A, respectively, were fixed). Jan. 13th version, minor changes, title change