Stable prescribed-mean-curvature integral varifolds of codimension 1: regularity and compactness

In a previous paper we developed a regularity and compactness theory in Euclidean ambient spaces for codimension 1 weakly stable CMC integral varifolds satisfying two (necessary) structural conditions. Here we generalize this theory to the setting where the mean curvature (of the regular part of the varifold) is prescribed by a given $C^{1, \alpha}$ ambient function $g$ and the ambient space is a general $(n+1)$-dimensional Riemannian manifold; we give general conditions that imply that the support of the varifold, away from a possible singular set of dimension $\leq n-7,$ is the image of a $C^{2}$ immersion with continuous unit normal $\nu$ and mean curvature $g\nu$. These conditions also identify a compact class of varifolds subject to an additional uniform mass bound. If $g$ does not vanish anywhere, or more generally if the set $\{g=0\}$ is sufficiently small (e.g. if ${\mathcal H}^{n}\,(\{g=0\})=0$), then the results and their proofs are quite analogous to the CMC case. When $g$ is arbitrary however, a number of additional considerations must be taken into account: first, the correct second variation assumption is that of finite Morse index (rather than weak stability); secondly, to have a geometrically useful theory with compactness conclusions, one of the two structural conditions on the varifold must be weakened; thirdly, in view of easy examples, this weakening of a structural hypothesis necessitates additional hypotheses in order to reach the same conclusion. We identify two sets of additional assumptions under which this conclusion holds for general $g$: one of them gives regularity and compactness, the other one gives regularity and is applicable to certain phase transition problems. We also provide corollaries for multiplicity $1$ varifolds associated to reduced boundaries of Caccioppoli sets. In these cases, some or all of the structural assumptions become redundant.