A remark on the existence of sphere with prescribed mean curvature (Q1383460)

In 1976, the author posed the following question: if \(\rho\) is a function defined on \(\mathbb{R}^3\), what is the condition on \(\rho\) so that there is a sphere in \(\mathbb{R}^3\) whose mean curvature is given by \(\rho\). This note gives a new partial solution to the above problem. Using his earlier paper (with R. Schoen) on the existence of black holes (see \textit{R. Schoen} and \textit{S.-T. Yau} [Commun. Math. Phys. 90, 575-579 (1983; Zbl 0541.53054)]), the author solves the prescribed mean curvature problem for a large class of \(\rho\).