The Boundary Yamabe Problem, II: General Constant Mean Curvature Case

TL;DR

This paper completely solves the boundary Yamabe problem with prescribed constant scalar and mean curvature on compact manifolds with boundary, using iterative schemes and perturbation methods, regardless of the Weyl tensor or boundary point classification.

Contribution

It introduces a new iterative scheme and perturbed conformal Laplacian approach to solve the boundary Yamabe problem in all cases, independent of dimension and boundary point classification.

Findings

Existence of solutions depends on the sign of the first eigenvalue of the conformal Laplacian.

The method applies to manifolds of any dimension without relying on Weyl tensor or boundary point classification.

A new monotone iteration scheme and gluing technique are developed for nonlinear elliptic PDEs with boundary conditions.

Abstract

This article uses the iterative schemes and perturbation methods to completely solve the Han-Li conjecture, i.e. the general boundary Yamabe problem with prescribed constant scalar curvature and constant mean curvature on compact manifolds $ (M, \partial M, g) $ with non-empty smooth boundary, $ \dim M \geqslant 3 $. It is equivalent to show the existence of a real, positive, smooth solution of $ -\frac{4(n -1)}{n - 2} \Delta_{g} u + R_{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 constants $ \lambda, \zeta \in \mathbb{R} $. This boundary Yamabe problem is solved in cases according to the sign of the first eigenvalue $ \eta_{1} $ of the conformal Laplacian with homogeneous Robin condition. The signs of scalar curvature $ R_{g} $ and mean curvature $ h_{g} $ play an important role in this existence result. In contrast to the classical method, the Weyl tensor and classification of boundary points play no role in the proof. In addition, the method is not dimensional specific. The key steps include a new version of monotone iteration scheme for nonlinear elliptic PDEs and nonlinear boundary conditions, an introduction of the perturbed conformal Laplacian, a delicate gluing skill to construct a smooth super-solution and existence of solution of local Yamabe-type equations.