CMC Foliations and their conformal aspects
arxiv(2023)
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.
MoreTranslated text
AI Read Science
Must-Reading Tree
Example
![](https://originalfileserver.aminer.cn/sys/aminer/pubs/mrt_preview.jpeg)
Generate MRT to find the research sequence of this paper
Chat Paper
Summary is being generated by the instructions you defined