CMC Foliations and their conformal aspects

Abstract

On a manifold we term a hypersurface foliation a slicing if it is the level set foliation of a slice function -- meaning some real valued function $f$ satisfying that $df$ is nowhere zero. On Riemannian manifolds we give a non-linear PDE on functions whose solutions are generic constant-mean-curvature (CMC) slice functions. Conversely, to any generic transversely-oriented constant-mean-curvature foliation the equation uniquely associates such a function. In one sense the equation is a scalar analogue of the Einstein equations. Given any slicing we show that, locally, one can conformally prescribe any smooth mean curvature function. We use this to show that, locally on a Riemannian manifold, a slicing is CMC for a conformally related metric. These results admit global versions assuming certain restrictions. Finally, given a conformally compact manifold we study the problem of normalising the defining function so that it is a CMC slice function for a compactifying metric. We show that two cases of this problem are formally solvable to all orders.