Foliations of a smooth metric measure space by hypersurfaces with constantf-mean curvature

Abstract

We study smooth codimension-one foliations F of a smooth metric measure space whose leaves have the same constant f -mean curvature. Firstly, we show that all the leaves of F are f -minimal hypersurfaces when either the smooth metric measure space is compact and has nonnegative Bakry‐ Emery Ricci curvature, or the limit of the ratio of the weighted volume of a geodesic ball B and the weighted area of a geodesic sphere @B vanishes. Secondly, we prove that every leaf of F is strongly f -stable. Lastly, we show that there is no complete proper foliation of the Gaussian space whose leaves have the same constant f -mean curvature. In particular, there are no foliations of R nC1 whose leaves are complete proper self-similar solutions for mean curvature flow.