# Ngô Quốc Anh

## May 11, 2011

### On the uniqueness for the Yamabe problem

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 14:25

The Yamabe problem has been discussed here. Basically, starting from a metric $g$, for a given constant $R$ Yamabe wanted to show there always exists a positive function $\varphi$ such that the scalar curvature of metric $\overline g$ defined to be $\varphi^\frac{4}{n-2}g$ equals $R$. In terms of PDEs, the scalar curvature satisfies the equation (called Yamabe equation)

$\displaystyle \frac{{4\left( {n - 1} \right)}} {{n - 2}}\Delta \varphi + R\varphi = \overline R {\varphi ^{\frac{{n + 2}} {{n - 2}}}}$

where $\overline R$ the scalar curvature of metric $\overline g$. It is not hard to see that in the negative and null cases, two solutions of the Yamabe equation with $\overline R = {\rm const}$ are proportional. Let us prove the following

Theorem. In the negative and null cases, two solutions of the Yamabe equation with $R=\overline R = {\rm const}$ are constant $1$.

We first fix a background metric $g_0$ and let $\varphi_0$ be a solution to the Yamabe equation with $\overline R=\mu$. That is, the scalar curvature of metric $\varphi_0^\frac{4}{n-2}g_0$ equals $\mu$.

We now consider a new metric $g_1$ still sitting in the same conformal class $[g_0]$ of $g_0$ with conformal factor $\varphi_1$ such that its scalar curvature equals $\mu$. In other words, $\varphi_1$ is also a solution of the Yamabe equation with $R=\overline R=\mu$. Keep in mind

$\displaystyle \frac{{4\left( {n - 1} \right)}} {{n - 2}}\Delta \varphi_1 + \mu\varphi_1 = \mu_1 {\varphi_1 ^{\frac{{n + 2}} {{n - 2}}}}.$

We have two cases.

• Suppose $\mu=0$. From the equation above, $\Delta \varphi_1=0$, that is, $\varphi_1$ is constant.
• Suppose $\mu<0$. At a point $P$ where $\varphi_1$ is maximum, $\Delta \varphi_1 \geqslant 0$, thus $\varphi_1(P) \leqslant 1$. Similarly, at a point $Q$ where $\varphi_1$ is minimum, $\Delta \varphi_1 \leqslant 0$, thus $\varphi_1(Q) \geqslant 1$. Consequently, $\varphi_1 \equiv 1$.

Notice that the uniqueness result in the positive  case is no longer true in general. Due to Obata, this is true for Einstein manifolds, that is, when the Ricci curvature and metric are proportional.