Prescribed mean curvature min-max theory in some non-compact manifolds

This paper develops a technique for applying one-parameter prescribed mean curvature min-max theory in certain non-compact manifolds. We give two main applications. First, fix a dimension 3 ≤ n + 1 ≤ 7 and consider a smooth function h : R n + 1 → R which is asymptotic to a positive constant near infinity. We show that, under certain additional assumptions on h , there exists a closed hypersurface Σ in R n + 1 with mean curvature prescribed by h . Second, let ( M 3 , g ) be an asymptotically flat 3-manifold with no boundary and fix a constant c > 0 . We show that, under an additional assumption on M , it is possible to find a closed surface Σ of constant mean curvature c in M .