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

Therefore, for the inverse metric, there holds

Since the normal vector field depends on the metric , it turns out that under the new metric ,

is a new normal vector field along , i.e. . To continue, we need to understand the conformal change for the Levi-Civita connection, let us recall the following formula

According to the note, the formula we quoted should be

provided . However, in view of our notation used, there holds . Hence

Then we calculate the new second fundamental form given by

Clearly, we have

Thus, we have shown that

Then

It is important to remind that the metric appearing in the above calculation is just the induced metric on which is of dimension , therefore is clear. In particular, when , we obtain

Therefore, if we normalize the mean curvature by

we then have the following rule

We now consider the case when the mean curvature is defined to be

Clearly, . Therefore, we can write

Thus, we have shown that

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