Ngô Quốc Anh

April 16, 2012

The Yamabe problem: The work by Neil Sidney Trudinger

Filed under: PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 2:34

Following the previous topic where we was able to point out the serious mistake in the Yamabe paper found by Trudinger. Today, we discuss about the way Trudinger did correct that mistake. Trudinger published the result in a paper entitlde “Remarks concerning the conformal deformation of Riemannian structures on compact manifolds” in Ann. Scuola Norm. Sup. Pisa in 1968. The paper can be downloaded from this link.

In the paper, he proved the following result

Theorem 2. There exists a positive constant $\varepsilon>0$ (depending on $g^{ij}$, $R$) such that if $\lambda<\varepsilon$, there exists a positive, $C^\infty$ solution of the equation $\displaystyle - \frac{{4(n - 1)}}{{n - 2}}{\Delta _g}\varphi + \underbrace {{\text{Scal}}_g}_R\varphi = \underbrace {{\text{Scal}}_{\widetilde g}}_{\widetilde R}{\varphi ^{\frac{{n + 2}}{{n - 2}}}},$

with $\widetilde R=\lambda$.

Let us discuss the proof of the above result. Again, the sub-critical approach was used in his argument and we refer the reader to the previous topic.

He said that we expect a subsequence of the $\varphi_q$ converges in a certain sense to a smooth solution of the critical equation. However, the convergence is not strong enough to imply the non-triviality of the resulting solution. Fortunately, if $\widetilde R$ is small enough, the convergence is sufficiently nice to guarantee a positive, smooth solution of the critical equation.

Recall that the function $\varphi_q$ verifies the sub-critical equation in the weak sense, that is, $\displaystyle \int_M \left(\frac{4(n-1)}{n-2} g^{ij}(\varphi_q)_i\xi_j + R\varphi_q \xi \right)dv= \mu_q\int_M\varphi_q^{q-1}\xi dv$

for all test functions $\xi \in H^1(M)$.

We now choose the test function $(\varphi_q)^\beta$ for some $\beta>1$. The result is $\displaystyle \int_M \left(\frac{4\beta (n-1)}{n-2} g^{ij}(\varphi_q)_i(\varphi_q)_j(\varphi_q)^{\beta-1} + R(\varphi_q)^{\beta+1}\right)dv= \mu_q\int_M(\varphi_q)^{\beta+q-1} dv$

and hence $\displaystyle\int_M {{{({\varphi _q})}^{\beta - 1}}|\nabla {\varphi _q}{|^2}dv} \leqslant \frac{{n - 2}}{{4\beta (n - 1)}}\int_M {[{\mu _q}{{({\varphi _q})}^{\beta + q - 1}} - R{{({\varphi _q})}^{\beta + 1}}]dv} .$

Writing $w=(\varphi_q)^\frac{\beta+1}{2}$, the above inequality becomes $\displaystyle\int_M {|\nabla w{|^2}dv} \leqslant \frac{{n - 2}}{{4\beta (n - 1)}}\int_M {[{\mu _q}{w^2}{{({\varphi _q})}^{q - 2}} - R{w^2}]dv} .$

Let us suppose $\lambda>0$. Using the Sobolev and Holder inequalities, we thus obtain $\displaystyle\left\| w \right\|_{{L^{{2^ \star }}}}^2 \leqslant {C_1}{\mu _q}\left\| w \right\|_{{L^{{2^ \star }}}}^2\left\| {{\varphi _q}} \right\|_{{L^{{2^ \star }}}}^2 + {C_2}\int_M {{w^2}dv} \leqslant {C_3}{\mu _q}\left\| w \right\|_{{L^{{2^ \star }}}}^2 + {C_2}\int_M {{w^2}dv}$

since $\left\| {{\varphi _q}} \right\|_{{L^{{2^ \star }}}}^2$ is uniformly bounded independent of $q$. Keep in mind that the constant $C_2$ depends on $\sup_M {\widetilde R}$ and $C_2>0$ if $\sup_M {\widetilde R}$ is sufficiently small. This is because of the second Sobolev constant in the Sobolev inequality in $H^1(M)$.

Hence, if $\mu_q<\frac{1}{C_3}$, for large $q$, we get that $\displaystyle\left\| w \right\|_{{L^{{2^ \star }}}}^2 \leqslant {C_4}\int_M {{w^2}dv}.$

Clearly, this estimate continues to hold if $\lambda \leqslant 0$. We now choose $\beta>0$ such that $\beta+1<2^\star$. With this, one can check that $\displaystyle\int_M {{w^2}dv} = \int_M {{{({\varphi _q})}^{\beta + 1}}dv} \leqslant \left\| {{\varphi _q}} \right\|_{{L^{\beta + 1}}}^{\beta + 1} \leqslant {C_5}\left\| {{\varphi _q}} \right\|_{{L^{{2^ \star }}}}^{\beta + 1},$

which implies that $\|w\|_{L^2}$ is bounded. This and the previous estimate tell us that $\|w\|_{L^{2^ \star}}$ is also bounded, that is, $\displaystyle\left\| w \right\|_{{L^{{2^ \star }}}} \leqslant C_6,$

which proves the equiboundedness of $\{\varphi_q\}_q$. Note that this is reasonable by standard elliptic theory since $\displaystyle\left\| w \right\|_{{L^{2^ \star }}} = \left\| {{\varphi _q}} \right\|_{{L^{\frac{{\beta + 1}}{2}{2^ \star }}}}$

and (this is the most important part) $\displaystyle {\frac{{\beta + 1}}{2}{2^ \star }}>2^\star.$

The proof completes.

Trudinger also proved a useful regularity result which we have discussed before, check this entry. As can be seen, for large $\lambda$, the solvablity of  the Yamabe was still left open. In the next entry, we talk about the work of Aubin.

1. Hi Ngo,
can you advice me a book or a survey on Yamabe problem?
txs

Comment by Fab — June 12, 2012 @ 15:10

• Just by myself, dude.

Comment by Ngô Quốc Anh — June 12, 2012 @ 22:59

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