The authors prove the existence of embedded spheres in \({\mathbb R}^3\) having prescribed mean curvature of the form \(H(u) = H_0 + \varepsilon H_1(u)\) where \(H_0\) is a non-zero constant, \(H_1:{\mathbb R}^3\to{\mathbb R}\) is a \(C^2\) function, and \(\varepsilon\) is of sufficiently small absolute value. Solutions are obtained as critical points of an appropriate energy functional. The finite-dimensional reduction method mentioned in the title refers to the fact that a set of solutions of the unperturbed problem defines a finite dimensional manifold \(Z\) of critical points for the unperturbed energy functional. The manifold \(Z\) is sufficiently well understood so that the authors can construct a \(3\)-dimensional manifold that serves as a natural constraint for the perturbed energy functional.
The final section of the paper contains a non-existence result.