An existence theorem of conformal scalar-flat metrics on manifolds with boundary

Abstract

Let (M, g) be a compact Riemannian manifold with boundary. We address the Yamabe-type problem of finding a conformal scalar-flat metric on M whose boundary is a constant mean curvature hypersurface. When the boundary is umbilic, we prove an existence theorem that finishes some of the remaining cases of this problem.

Harmonically free group actions and equivariant bifurcation in the Yamabe problem

“…The existence of solution for these problems was proved in the works of Cherrier [9], Escobar [12], [13], [14], Almaraz [1], Han [15], Marques [17], [18] and others.…”

“…that is an embedded submanifold of M k,α (M ) 1 and an embedded submanifold of [g]. This conformal class can be expressed as…”

“…, consisting of eigenfunctions of ∆ g (1) and of ∆ g (2) , respectively. We (2) are eigenfunctions of ∆ g (1) , ∆ g (2) , associated to the eigenvalues ρ (1) , ρ (2) , respectively, then f (1) ⊗ f (2) is an eigenfunction of ∆ g associated to the eigenvalue ρ (1) + ρ (2) .…”

“…Let (M 1 , g (1) ) be a compact Riemannian manifold, with ∂M 1 = ∅ and constant scalar curvature, and let (M 2 , g (2) ) be a compact Riemannian manifold with minimal boundary and constant scalar curvature. Consider the product manifold, M = M 1 × M 2 , which boundary is given by ∂M = M 1 × ∂M 2 .…”

“…In [10] we studied multiplicity of solutions of the the Yamabe problem, according to formulation (i) above, in a manifold (M, g s ) obtained as the product of a compact Riemannian manifold (M 1 , g (1) ) (without boundary) with constant scalar curvature and a compact Riemannian manifold (M 2 , g (2) ) with constant scalar curvature and minimal boundary, where M = M 1 × M 2 , ∂M = M 1 × ∂M 2 = ∅, and g s = g (1) ⊕ sg (2) is a family of metrics of constant scalar curvature and vanishing mean curvature of the boundary, for all s ∈ (0, ∞).…”