Conformal Deformation on Manifolds with Boundary

We consider natural conformal invariants arising from the Gauss-Bonnet formulas on manifolds with boundary, and study conformal deformation problems associated to them. The purpose of this paper is to study conformal deformation problems associated to conformal invariants on manifolds with boundary. From analysis point of view, the problem becomes a non-Dirichlet boundary value problems for fully nonlinear equations. This may be compared to a work by Lieberman-Trudinger [22] on the oblique-type boundary value problems. Let (M, g) be a compact, connected Riemannian manifold of dimension n ≥ 3 with boundary ∂M . We denote the Riemannian curvature, Ricci curvature, scalar curvature, mean curvature, and the second fundamental form by Riem,Ric, R, h, and Lαβ , respectively. The Yamabe constant for compact manifolds with boundary is a conformal invariant, defined as Y (M, ∂M, [g]) = inf ĝ∈[g],Vĝ=1 ( ∫