Ngô Quốc Anh

September 25, 2009

An introduction to Yamabe problem

Filed under: Nghiên Cứu Khoa Học, PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 15:51

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

On a compact Riemannian manifold $(M, g)$, there exists a metric $g'$ conformal to $g$, such that the corresponding scalar curvature $R'$ is constant”.

The Yamabe problem was born, since there is a gap in Yamabe’s proof. Now there are many proofs of this statement.

Let us recall the question. Let $(M_n, g)$ be a compact $C^\infty$-Riemannian manifold of dimension $n \geq 3$, is its scalar curvature. The problem is:

Does there exists a metric $g'$, conformal to $g$, such that the scalar curvature $R'$ of the metric $g$ is constant?”.

Let us consider the conformal metric $g'=e^fg$ with $f \in C^\infty$. If $\Gamma'^l_{ij}$ and $\Gamma_{ij}^l$ denote the Chrisoffel symbols relating to $g'$ and $g$, respectively, then

$\displaystyle\Gamma '^l_{ij} - \Gamma _{ij}^l = \frac{1} {2}\left( {{g_{kj}}\frac{{\partial f}} {{\partial {x_i}}} + {g_{ki}}\frac{{\partial f}} {{\partial {x_j}}} - {g_{ij}}\frac{{\partial f}} {{\partial {x_k}}}} \right){g^{kl}} = \frac{1} {2}\left( {\delta _j^l{\partial _i}f + \delta _i^l{\partial _j}f - {g_{ij}}{\nabla ^l}f} \right).$

Clearly,

$\displaystyle R'_{ij}=R'^k_{ikj}=R_{ij}-\frac{n-2}{2}\nabla _{ij}f+\frac{n-2}{4}\nabla _if\nabla _jf-\frac{1} {2}\left(\nabla _\nu^\nu f+\frac{n-2}{2}\nabla^\nu f\nabla _\nu f\right)g_{ij}$

so

$\displaystyle R' = {e^{ - f}}\left( {R - \left( {n - 1} \right)\nabla _\nu ^\nu f - \frac{{\left( {n - 1} \right)\left( {n - 2} \right)}} {4}{\nabla ^\nu }f{\nabla _\nu }f} \right).$

If we consider the conformal deformation in the form $g'=\varphi^\frac{4}{n-2}g$ (with $\varphi \in C^\infty$, $\varphi>0$), the scalar curvature satisfies the equation

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

where $\Delta \varphi = - {\nabla ^\nu }{\nabla _\nu }\varphi$. So, Yamabe problem is equivalent to solving the above equation with $R'=const$ and the solution $\varphi$ must be smooth and strictly positive.

Link to PDF file of the paper Osaka Math. J. 12 (1960), pp. 21-37 can be found here http://projecteuclid.org/euclid.ojm/1200689814