# Ngô Quốc Anh

## March 10, 2011

### log K is harmonic implies that K is contant

Filed under: PDEs — Ngô Quốc Anh @ 18:29

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 $(\mathbb S^2,g_0)$ with standard metric, if $K(x)>0$ and $\Delta \ln \big(K(x)\big)=0$, then $K$ must be a constant.

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

$\displaystyle \Delta \ln K=\frac{\Delta K}{K}-\frac{|\nabla K|^2}{K^2}.$

Therefore, the fact that $\ln K$ is harmonic implies that

$\displaystyle\frac{\Delta K}{K}=\frac{|\nabla K|^2}{K^2}.$

Since $K>0$, multiplying both sides by $K^2$, we get

$\displaystyle K\Delta K=|\nabla K|^2.$

Integrating both sides on $\mathbb S^2$ using standard volume form, we get

$\displaystyle \int_{\mathbb S^2}K\Delta K dv_{g_0}=\int_{\mathbb S^2}|\nabla K|^2dv_{g_0}.$

Now integration by parts shows that

$\displaystyle -\int_{\mathbb S^2}|\nabla K|^2dv_{g_0}=\int_{\mathbb S^2}|\nabla K|^2dv_{g_0}.$

This in turn shows that

$|\nabla K|=0.$

We conclude that $K(x)$ must be a constant function.