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

Abstract

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 $3\le n+1\le 7$ and consider a smooth function $h:R^{n+1}\to R$ 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.