Ngô Quốc Anh

June 28, 2011

The Yamabe problem: A Story

Filed under: PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 15:47

I want to write a short survey about the Yamabe problem. Long time ago, I introduced the problem in this blog [here] but it turns out that the note was not rich enough to perform the importance of the problem.

Hidehiko Yamabe, in his famous paper entitled On a deformation of Riemannian structures on compact manifolds, Osaka Math. J. 12 (1960), pp. 21-37,  wanted to solve the Poincaré conjecture

Conjecture. Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere

For this he thought, as a first step, to exhibit a metric with constant scalar curvature. We refer the reader to this note for details. He considered conformal metrics (the simplest change of metric is a conformal one), and gave a proof of the following statement:

Theorem (Yamabe). On a compact Riemannian manifold (M, g) of dimension \geqslant 3, there exists a metric g' conformal to g, such that the corresponding scalar curvature \text{Scal}_{g'} is constant.

As can be seen, the Yamabe problem is a special case of the prescribing scalar curvature problem that can be completely solved. For the prescribing scalar curvature, we also solve it completely when the invariant is non-positive.

1. Conformal metrics.

Definition (conformal). Two pseudo-Riemannian metrics g and \widetilde g on a manifold M are said to be

  • (pointwise) conformal if there exists a C^\infty function f on M such that

    \displaystyle \widetilde g=e^{2f}g;

  • conformally equivalent if there exists a diffeomorphism \alpha of M such that \alpha^* \widetilde g and g are pointwise conformal.

Note that, if g and \widetilde g are conformally equivalent, then \alpha is an isometry from e^{2f}g onto \widetilde g. So we will only study below the case \widetilde g = e^{2f}g.

2. Scalar curvature under conformal changes of Riemannian metrics.

We have already shown in this entry that under the conformal change \widetilde g = e^{2f} g, the scalar curvatures verify the following equation

\displaystyle{\rm Scal}_{\widetilde g} = {e^{ - 2f}}\left[ {{\rm Scal}_g - 2(n - 1)\Delta_g f- (n - 2)(n - 1)|{\rm grad} f{|_g^2}} \right].

If we consider the conformal deformation in the form \widetilde g=\varphi^\frac{4}{n-2}g (with \varphi \in C^\infty, \varphi>0), that is,

\displaystyle e^{2f}=\varphi^\frac{4}{n-2},

we then have

\displaystyle {e^f}{\partial _i}f = \frac{2}{{n - 2}}{\varphi ^{ - \frac{{n - 4}}{{n - 2}}}}{\partial _i}\varphi.

In other words,

\displaystyle {\partial _i}f = \frac{2}{{n - 2}}{\varphi ^{ - 1}}{\partial _i}\varphi.

Thus

\displaystyle\left| \text{grad}f \right|_g^2 = {g^{ij}}{\partial _i}f{\partial _j}f = {\left( {\frac{2}{{n - 2}}} \right)^2}{\varphi ^{ - 2}}{g^{ij}}{\partial _i}\varphi {\partial _j}\varphi

and (see this note for a similar calculation for Laplacian)

\displaystyle {\Delta _g}f = - \frac{2}{{n - 2}}{\varphi ^{ - 2}}{g^{ij}}{\partial _i}\varphi {\partial _j}\varphi + \varphi^{-1}{\Delta _g}\varphi .

Therefore,

\displaystyle \begin{gathered} {\text{Sca}}{{\text{l}}_{g}} - 2(n - 1){\Delta _g}f - (n - 2)(n - 1)|{\text{grad}}f|_g^2 \hfill \\\qquad= {\text{Sca}}{{\text{l}}_g} + 4\frac{{n - 1}}{{n - 2}}{\varphi ^{ - 2}}{g^{ij}}{\partial _i}\varphi {\partial _j}\varphi - 4\frac{{n - 1}}{{n - 2}}{\varphi ^{ - 1}}{\Delta _g}\varphi \hfill \\ \qquad- (n - 2)(n - 1){\left( {\frac{2}{{n - 2}}} \right)^2}{\varphi ^{ - 2}}{g^{ij}}{\partial _i}\varphi {\partial _j}\varphi \hfill \\\qquad = {\text{Sca}}{{\text{l}}_g} - 4\frac{{n - 1}}{{n - 2}}{\varphi ^{ - 1}}{\Delta _g}\varphi . \hfill \\ \end{gathered}

Hence,

\displaystyle {\text{Sca}}{{\text{l}}_{\widetilde g}} = {\varphi ^{ - \frac{4}{{n - 2}}}}\left( {{\text{Sca}}{{\text{l}}_g} - 4\frac{{n - 1}}{{n - 2}}{\varphi ^{ - 1}}{\Delta _g}\varphi } \right).

3. The Yamabe approach.

From the previous section, the scalar curvature satisfies the equation

\displaystyle - 4\frac{{n - 1}}{{n - 2}}{\Delta _g}\varphi + {\text{Sca}}{{\text{l}}_g}\varphi = {\text{Sca}}{{\text{l}}_{\widetilde g}}{\varphi ^{\frac{{n + 2}}{{n - 2}}}}.

So, Yamabe problem is equivalent to solving the above equation with R'=\text{const} and the solution \varphi must be smooth and strictly positive.

4. The mistake and…

The Yamabe problem was born, since there is a gap in Yamabe’s proof.

See also: Conformal invariant operators: Laplacian operators, 2

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