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