The Han-Li conjecture in constant scalar curvature and constant boundary mean curvature problem on compact manifolds

Abstract

The Han-Li conjecture states that: Let (M,g0) be an n-dimensional (n≥3) smooth compact Riemannian manifold with boundary having positive (generalized) Yamabe constant and c be any real number, then there exists a conformal metric of g0 with scalar curvature 1 and boundary mean curvature c. Combining with Z.C. Han and Y.Y. Li's results, we answer this conjecture affirmatively except for the case that n≥8, the boundary is umbilic, the Weyl tensor of M vanishes on the boundary and has an interior non-zero point.