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