Ngô Quốc Anh

March 20, 2012

The Yamabe problem: The work by Hidehiko Yamabe

Filed under: PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 20:58

Following the previous post, we are interested in solving the following equation $\displaystyle - 4\frac{{n - 1}}{{n - 2}}{\Delta _g}\varphi + {\text{Sca}}{{\text{l}}_g}\varphi = {\text{Sca}}{{\text{l}}_{\widetilde g}}{\varphi ^{\frac{{n + 2}}{{n - 2}}}},$

where $\widetilde g=\varphi^\frac{4}{n-2}g$ (with $\varphi \in C^\infty$, $\varphi>0$) is a conformal metric conformally to $g$. In this entry, we introduce the Hidehiko Yamabe approach. His approach is variational. To keep his notation used, we rewrite the PDE as the following $\displaystyle -\Delta \varphi + R\varphi = C_0 \varphi^\frac{n+2}{n-2}.$

Yamabe tried to minimize the following $\displaystyle {F_q}(u) = \frac{{\displaystyle\int_M {\left( {\frac{{4(n - 1)}}{{n - 2}}|\nabla u{|^2} + R{u^2}} \right)d{v_g}} }}{{{{\left( {\displaystyle\int_M {|u{|^q}d{v_g}} } \right)}^{\frac{2}{q}}}}}$

over the Sobolev space $H^1(M)$ where $q \leqslant \frac{2n}{n-2}$. Let us say $\displaystyle {\mu _q} = \mathop {\inf }\limits_{u \in {H^1}(M)} {F_q}(u).$

In the first stage, he showed that

Theorem B. For any $q<\frac{2n}{n-2}$, there exists a positive function $\varphi_q$ satisfying $\displaystyle -\Delta \varphi_q + R\varphi_q = \mu_q \varphi_q^\frac{n+2}{n-2}.$

In the second stage, he sent $q$ to $\frac{2n}{n-2}$ to obtain a solution. He proved

Theorem C. As $q$ tends to $\frac{2n}{n-2}$, a uniform limit $\varphi$ of such $\varphi_q$ exists, is positive and satisfies $\displaystyle -\Delta \varphi + R\varphi = \mu_\frac{2n}{n-2} \varphi^\frac{n+2}{n-2}.$

Once the preceding PDE is solved, one can replace $\mu_\frac{2n}{n-2}$ by any constant, thus, finishing the prescribing scalar curvature in the constant case.

While the proof of Theorem B is correct, a mistake was found in the proof of Theorem C by Trudinger. As can be expected, in order to send $q$ to $\frac{2n}{n-2}$, one needs to derive some uniform bound for the sequence of solutions of subcritical equations. In order to do that, Yamabe used the Green function $G(P,Q)$ for the operator $\displaystyle \frac{4(n-1)}{n-2}\Delta.$

It is important to note that if $u \in L^{q'}$ and we define $\displaystyle u_1(P) = \int_M G(P,Q)u(Q)dv(Q)$

then

1. $u_1(P) \in L^{q'}$ for any $q'$ satisfying $\displaystyle \frac{1}{p'} \geqslant \frac{1}{q'}-\frac{2}{n}.$

2. Therefore $\|u_1||_{p'}=C_5\|u\|_{q'}$ where $C_5$ is an absolute constant if $\frac{1}{p'}-\frac{1}{q'}+\frac{2}{n}$ is larger than a fixed constant.

Having this result, Yamabe used an iteration to derive an uniform boundedness. Take a positive fixed $\xi_2\in (0,1)$. Starting at $\displaystyle q_1=\frac{2n}{n-2}+\xi_2,$

he then defined $\displaystyle q_k=\frac{2n}{n-2}+\left(\frac{n+2}{n-2}\right)^{k-1}\xi_2, \quad k \geqslant 2.$

It is immediately to check that $\displaystyle {q_{k + 1}}{q_k} = \frac{{4{n^2}}}{{{{(n - 2)}^2}}} + \frac{{4{n^2}}}{{{{(n - 2)}^2}}}{\left( {\frac{{n + 2}}{{n - 2}}} \right)^{k - 1}}{\xi _2} + {\left( {\frac{{n + 2}}{{n - 2}}} \right)^{2k - 1}}\xi _2^2.$

Therefore, $\displaystyle\begin{gathered} \frac{1}{{{q_{k + 1}}}} - \frac{1}{{{q_k}}}\frac{{n + 2}}{{n - 2}} + \frac{2}{n} = \frac{{(n - 2){q_k} - (n + 2){q_k}}}{{{q_{k + 1}}{q_k}}}\frac{1}{{n - 2}} + \frac{2}{n} \hfill \\ \qquad\qquad= \frac{{2n + \frac{{{{(n + 2)}^{k - 1}}}}{{{{(n - 2)}^{k - 2}}}}{\xi _2} - \frac{{2n(n + 2)}}{{n - 2}} - \frac{{{{(n + 2)}^{k + 1}}}}{{{{(n - 2)}^k}}}{\xi _2}}}{{{q_{k + 1}}{q_k}}}\frac{1}{{n - 2}} + \frac{2}{n} \hfill \\ \qquad\qquad= \frac{{ - \frac{{8n}}{{n - 2}}\left( {1 + {{\left( {\frac{{n + 2}}{{n - 2}}} \right)}^{k - 1}}{\xi _2}} \right)}}{{{q_{k + 1}}{q_k}}}\frac{1}{{n - 2}} + \frac{2}{n} \hfill \\ \qquad\qquad= \frac{2}{n}\frac{1}{{{q_{k + 1}}{q_k}}}\left[ {{q_{k + 1}}{q_k} - \left( {1 + {{\left( {\frac{{n + 2}}{{n - 2}}} \right)}^{k - 1}}{\xi _2}} \right)\frac{{4{n^2}}}{{{{(n - 2)}^2}}}} \right] \hfill \\ \qquad\qquad= \frac{2}{n}\frac{1}{{{q_{k + 1}}{q_k}}}{\left( {\frac{{n + 2}}{{n - 2}}} \right)^{2k - 1}}\xi _2^2. \hfill \\ \end{gathered}$

Clearly, using the Green function, there holds $\displaystyle {\varphi _q}(P) = - \int {G(P,Q)[ - {\mu _q}(\varphi _q^{q - 1} + R{\varphi _q})]dv(Q)} + \int {{\varphi _q}(Q)dv(Q)} .$

Therefore, due to the presence of the high order term, if $\varphi_q \in L^{q_k}$, we would have $\displaystyle \varphi_q^{q-1} \in L^\frac{q_k}{q-1}.$

Using this, we get that $\displaystyle \|\varphi_q\|_{q_{k+1}} \leqslant C_5 \|\varphi_q^{q-1}\|_\frac{q_k}{q-1}.$

Here we have used the following fact $\displaystyle\frac{1}{{{q_{k + 1}}}} - \frac{{q - 1}}{{{q_k}}} + \frac{2}{n} > \frac{1}{{{q_{k + 1}}}} - \frac{1}{{{q_k}}}\frac{{n + 2}}{{n - 2}} + \frac{2}{n} > 0$

to get the estimate. Clearly, $\displaystyle\|\varphi_q^{q-1}\|_\frac{q_k}{q-1}=\|\varphi_q\|_{q_k}^{q-1}.$

In other words, we would have $\displaystyle \|\varphi_q\|_{q_{k+1}} \leqslant C_5\|\varphi_q\|_{q_k}^{q-1}.$

If we repeat this procedure, we would have $\displaystyle \|\varphi_q\|_{q_{k+1}} \leqslant C_5\|\varphi_q\|_{q_1}^{(q-1)^k}.$

Unfortunately, this estimate was not calculated correctly in the Yamabe paper. To be precise, he claimed that $\displaystyle \|\varphi_q\|_{q_{k+1}} \leqslant C_5^{n-1}\|\varphi_q\|_{q_1}.$

This points out a serious mistake found by Trudinger. In other words, in view of the iteration procedure, to compare norms between $L^{p_{k+1}}$ and $L^{p_k}$, we first compare norms between $L^{p_{k+1}}$ and $L^\frac{p_k}{p-1}$ and then compare norms between $L^\frac{p_k}{p-1}$ and $L^{p_k}$. It turns out that Yamabe made a mistake in the latter part where he missed to raise to power $p-1$.

Now let us see the rest of Yamabe’s argument if this mistake were correct. Indeed, by denoting $\Lambda (q)=\sup_P\varphi_q(P),$

we get by the Holder inequality, since $\frac{q_k-1}{q_k} +\frac{1}{q_k}=1$, that $\displaystyle\Lambda (q) \leqslant \sup_P \|G(P,Q)\|_\frac{q_k}{q_k-1} \|\varphi_q\|_{q_k} \leqslant C_6\|\varphi_q\|_{q_1}.$

However, from the choice of $\Lambda$, we get $\displaystyle\|\varphi_q\|_{q_1}^{q_1} = \int_M \varphi_q^{q_1} = \int_M \varphi_q^q \varphi_q^{q_1-q}\leqslant \Lambda(q)^{q_1-q}\|\varphi_q\|_q^q=\Lambda(q)^{q_1-q}=\Lambda(q)^{\xi_2 + \varepsilon}.$

Hence, for $\varepsilon>0$ small and since $q_1>1$, there holds $\displaystyle \|\varphi_q\|_{q_1} \leqslant\Lambda(q)^\frac{\xi_2 + \varepsilon}{q_1} \leqslant \max\{\Lambda (q)^{\xi_2},1\}.$

Combining all, we arrive at $\displaystyle \Lambda (q) \leqslant\max\{\Lambda (q)^{\xi_2},1\}$

which immediately implies that $\Lambda(q)$ is bounded since $\xi_2\in(0,1)$. Notice that if we presume that Yamabe did correct, then we can go though the proof of uniformly boundedness of the sequence of solutions $\{\varphi_q\}_q$ since the bound for $\Lambda(q)$ is independent of $q$ so that the sequence of solutions $\{\varphi_q\}_q$ is uniformly bounded.

If we modify the above argument using the correct estimate, we would have $\displaystyle\Lambda (q) \leqslant \sup_P \|G(P,Q)\|_\frac{q_k}{q_k-1} \|\varphi_q\|_{q_k} \leqslant C_6\|\varphi_q\|_{q_1}^{(q-1)^k}.$

But then we get $\Lambda (q) \leqslant C_6\|\varphi_q\|_{q_1}^{(q-1)^k} \leqslant C_6\Lambda(q)^{\frac{\xi_2 + \varepsilon}{q_1}(q-1)^k},$

which gives us nothing since the exponent $\frac{\xi_2 + \varepsilon}{q_1}(q-1)^k$ is too big, in fact, converging to $+\infty$ as $q \to +\infty$ and we have no way to control from above, even for the case $k=1$ when no iteration happens.

We close this note by quoting the following in a paper by Trudinger: “More intuitively speaking, his proof of Theorem C cannot be expected to work since it does not distinguish at all two facts related to the problem

1. the presence of the term $-Ru$ in the equation and
2. the compactness of $M$“.