  We introduce an iterative scheme to solve the Yamabe equation $ - a\Delta_{g}
u + S u = \lambda u^{p-1} $ on small domains $(\Omega,g)\subset {\mathbb R}^n$
equipped with a Riemannian metric $g$. Thus $g$ admits a conformal change to a
constant scalar curvature metric. The proof does not use the traditional
functional minimization. Applications to the Yamabe problem on closed
manifolds, manifolds with boundary, and noncompact manifolds are given in
forthcoming papers.
