In a previous post, I showed how the mean curvature changes under a conformal change using slightly local coordinates approach. Today, I want to reconsider that topic using global approach.
As usual, suppose is an
-dimensional Riemannian manifold with boundary
. We also assume that
is an outward unit normal vector field along the boundary
. By an unit normal vector field we mean
and
for any tangent vectors
of
. Note that
is just a hypersurface of
and we also use
to denote the induced metric of
onto
. Then we have the so-called second fundamental form
associated to
defined to be
for any tangent vectors of
.
Regarding to the mean curvature , we shall use the following definition
Our aim is to calculate the mean curvature under the following conformal change
for some smooth positive function
and a real number
. It is important to note that by
we mean