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 $3\le n+1\le 7$ and consider a smooth function $h\text{\hspace{0.17em}⁣}:{\mathbb{R}}^{n+1}\to \mathbb{R}$ which is asymptotic to a positive constant near infinity. We show that, under certain additional assumptions on $h$, there exists a closed hypersurface $\mathrm{\Sigma }$ in ${\mathbb{R}}^{n+1}$ with mean curvature prescribed by $h$. Second, let $(M^3,g)$ be an asymptotically flat 3-manifold and fix a constant $c>0$. We show that, under an additional assumption on $M$, it is possible to find a closed surface $\mathrm{\Sigma }$ of constant mean curvature $c$ in $M$.