Ngô Quốc Anh

November 2, 2013

The Yamabe problem: The work by Richard Melvin Schoen

Filed under: PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 3:52

By using the notations used in the note about the work of Yamabe, they are

\displaystyle {F_q}(u) = \frac{{\displaystyle\int_M {\left( {\frac{{4(n - 1)}}{{n - 2}}|\nabla u{|^2} + R{u^2}} \right)d{v_g}} }}{{{{\left( {\displaystyle\int_M {|u{|^q}d{v_g}} } \right)}^{2/q}}}}

where q \leqslant \frac{2n}{n-2} and

\displaystyle {\mu _q} = \mathop {\inf }\limits_{u \in {H^1}(M)} {F_q}(u),

For simplicity, we set

\displaystyle E(u) = \int_M {\left( {|\nabla u{|^2} + \frac{{n - 2}}{{4(n - 1)}}R{u^2}} \right)d{v_g}} .

Schoen proved that:

Theorem. In any case, there holds

\displaystyle\mu_{2n/(n-2)} \leqslant n(n-1)\omega_n^{2/n},

and the equality occurs if and only if M is conformally equivalent to the sphere with standard metric.

In order to prove the above result, it suffices to consider the case either n=3,4,5 or M is locally conformally flat at some point. The main ingredient of his proof is the positive mass theorem. This well-known result says that if the metric g of M is conformally flat in a neighborhood of O, then the Green fucntion of the conformal Laplacian

 \displaystyle Lu=\Delta u - \frac{n-2}{4(n-1)}Ru,

say G, has an expansion  in suitable coordinates  near O as follows

\displaystyle G(x)=|x|^{2-n}+A+O(|x|)

with A \geqslant 0. Moreover, A=0 if and only if M is isometric to the standard flat \mathbb R^n. In the case M is locally conformally flat at some point, the positive mass theorem says that A >0.

Instead of using the test functions

\displaystyle \varphi_\varepsilon (x) = \left( \frac{\varepsilon}{\varepsilon^2+|x|^2}\right)^{(n-2)/2}

he used a small multiple of G outside a neigborhood of O, say \varphi, verifies

\displaystyle F_{2n/(n-2)}(\varphi) < n(n-1)\omega_n^{2/n}.

Precisely, he set

\displaystyle\varphi (x) = \left\{ \begin{gathered} {\varphi _\varepsilon }(x), \qquad\qquad\qquad\qquad|x| \leqslant {\rho _0}, \hfill \\ {\varepsilon _0}(G(x) - \psi (x)\alpha (x)), \quad {\rho _0} \leqslant |x| \leqslant 2{\rho _0}, \hfill \\ {\varepsilon _0}G(x), \qquad\qquad\qquad\qquad 2{\rho _0} \leqslant |x|, \hfill \\ \end{gathered} \right.

for some suitable \varepsilon, \psi (x) in such a way that \varphi is continuous across the boundary \partial B_{\rho_0}(0). Then after several long calculation but worth-doing, he realized that

\displaystyle E(\varphi ) \leqslant {\mu _{2n/(n - 2)}}{\left( {\int_{{B_{{\rho _0}}}} {{\varphi ^{2n/(n - 2)}}d{v_g}} } \right)^{(n - 2)/n}} - (n - 2){\sigma _{n - 2}}A\varepsilon _0^2 + c\rho _0^{ - n}\varepsilon _0^{1 + n/(n - 2)} + c{\rho _0}\varepsilon _0^2,

where \sigma_{n-2} denotes the volume of \mathbb S^{n-1}. When M is a conformally flat manifold, A>0. Hence, by choosing \rho_0 small and \varepsilon_0 much smaller than \rho_0, we obtain

\displaystyle - (n - 2){\sigma _{n - 2}}A\varepsilon _0^2 + c\rho _0^{ - n}\varepsilon _0^{1 + n/(n - 2)} + c{\rho _0}\varepsilon _0^2 < 0

showing that \mu_{2n/(n-2)}<n(n-1)\omega_n^{2/n} as we expected. A careful study shows that in the case n=3, the Green function still has the same expansion as showed above for the locally conformally flat case. Therefore, the above argument works in the case n=3 as well.

For the case n=4,5, Schoen used a modified metric {}^\rho g which agrees with g outside a small ball centered in O and Euclidean in that small ball. When |x| is small, the corresponding Green function for {}^\rho g has the following expansion

\displaystyle G_\rho (x)=|x|^{2-n}+A_\rho+O(|x|).

Schoen showed that one can conclude the theorem provided

\displaystyle\liminf_{\rho \to 0} A_\rho >0.

This turns out to be the key point since we don’t have any positive mass theorem in hand. Nevertheless, Schoen proved that this is the case provided the original (M,g) is not conformally isometric to \mathbb R^n.

Thus, in the case n=4,5, we start with a modified metric instead of the original one and look for a metric conformally to the modified one to get the positive constant scalar curvature. Once we can conclude

\displaystyle\mu_{2n/(n-2)} < n(n-1)\omega_n^{2/n}

in most of cases mentioned above, we can use the method of variation to complete the proof. More precise, having this estimate, we can get an uniform lower bound on the L^2 norm of the sequence of solutions of subcritical problems. Thus, this procedure generates a non-trivial solution of the critical problem.

Just a very quick remark, the sequence of solutions of subcritical problems always converges to a solution of the critical problem. Unfortunately, there is no reason to guarantee that this limiting solution is non-trivial. I have touched this part of argument in my recent paper published in Advances in Mathematics. In my case, we can have an uniform lower bound on the sequence of solutions of subcritical problems, thanks to the term with an negative exponent. In the Yamabe problem, the key argument to make sure that the limiting solution is non-trivial is to prove that

\displaystyle\mu_{2n/(n-2)} < n(n-1)\omega_n^{2/n}.

See also:

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