Closed surface with prescribed mean curvature in \(\mathbb{R}^ 3\) (Q1180008)

An interesting problem in geometry is to find conditions such that a given function \(H\) defined on \(\mathbb{R}^ 3\) is the mean curvature of a smooth closed embedded surface \(M\). One strategy is to take a primary surface \(M_ 0\) in \(\mathbb{R}^ 3\) and seek a surface \(M\) that can be represented as a graph over \(M_ 0\). This representation of \(M\) leads to a nonlinear second order elliptic partial differential equation for a function \(u\) on \(M_ 0\) -- a mean curvature type equation with prescribed mean curvature \(H\). The author succeeds in solving this equation, and consequently in finding a surface \(M\) as above with the same genus as \(M_ 0\), under two conditions on \(M_ 0\) and \(H\). The first of these is that a parallel surface of \(M_ 0\) on each side of \(M_ 0\) should be a barrier for \(u\). The second condition is more technical -- it is used to obtain an a priori gradient bound for \(u\). Results for the special case that \(M_ 0\) is a sphere were previously obtained by \textit{I. Ya. Bakel'man} and \textit{B. E. Kantor} [Geom. Topol. 1, 3-10 (1974)] and \textit{A. E. Treibergs} and \textit{S. W. Wei} [J. Differ. Geom. 18, 513-521 (1983; Zbl 0529.53043)].