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“.

See also:

  • The Yamabe problem: A Story 
  • The Yamabe problem: The work by Hidehiko Yamabe
  • The Yamabe problem: The work by Neil Sidney Trudinger
  • The Yamabe problem: The work by Thierry Aubin
  • The Yamabe problem: The work by Richard Melvin Schoen

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