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.

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