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In a previous paper we developed a regularity and compactness theory inEuclidean ambient spaces for codimension 1 weakly stable CMC integral varifoldssatisfying two (necessary) structural conditions. Here we generalize thistheory to the setting where the mean curvature (of the regular part of thevarifold) is prescribed by a given $C^{1, \alpha}$ ambient function $g$ and theambient space is a general $(n+1)$-dimensional Riemannian manifold; we givegeneral conditions that imply that the support of the varifold, away from apossible singular set of dimension $\leq n-7,$ is the image of a $C^{2}$immersion with continuous unit normal $\nu$ and mean curvature $g\nu$. Theseconditions also identify a compact class of varifolds subject to an additionaluniform mass bound.  If $g$ does not vanish anywhere, or more generally if the set $\{g=0\}$ issufficiently small (e.g. if ${\mathcal H}^{n}\,(\{g=0\})=0$), then the resultsand their proofs are quite analogous to the CMC case. When $g$ is arbitraryhowever, 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 theorywith compactness conclusions, one of the two structural conditions on thevarifold must be weakened; thirdly, in view of easy examples, this weakening ofa structural hypothesis necessitates additional hypotheses in order to reachthe same conclusion. We identify two sets of additional assumptions under whichthis conclusion holds for general $g$: one of them gives regularity andcompactness, the other one gives regularity and is applicable to certain phasetransition problems. We also provide corollaries for multiplicity $1$ varifoldsassociated to reduced boundaries of Caccioppoli sets. In these cases, some orall of the structural assumptions become redundant.

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