In this short note we present a result in a paper due to Edward M. Fan [here]. To be precise, we prove

Given a sphere with standard metric, if and , then must be a constant.

*Proof*. To prove the result, we shall use the following well-known formula

Therefore, the fact that is harmonic implies that

Since , multiplying both sides by , we get

Integrating both sides on using standard volume form, we get

Now integration by parts shows that

This in turn shows that

We conclude that must be a constant function.

### Like this:

Like Loading...

*Related*

Where exactly does the proof go wrong when considering manifolds which are not the sphere with the round metric?

Comment by OrbiculaR — March 11, 2011 @ 1:05

Hi, thanks for your interest in my blog. Apparently, the conclusion still holds as long as the manifolds are closed in the sense that they are compact without boundary.

Comment by Ngô Quốc Anh — March 11, 2011 @ 2:25