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).

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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}.

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November 8, 2011

A blowup proof of the Aubin theorem in the Yamabe problem

Filed under: PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 13:29

Yamabe’s approach was to consider first the perturbed functional

\displaystyle Q_s(u)\doteqdot\frac{\displaystyle\int_M\Big(|\nabla u|^2+\frac{n-2}{4(n-1)}R_gu^2\Big)d\mu_g}{\left(\displaystyle\int_M|u|^sd\mu_g\right)^\frac{2}{s}}

where

\displaystyle s\in \left(0,\frac{2n}{n-2} \right] \quad \text{ and } \quad u\in H^1(M)\setminus\{0\}.

Set

\displaystyle \lambda_s\doteqdot\inf\big\{Q_s(u):u\in H^{1}(M)\setminus\{0\}\big\}\quad\text{and}\quad Y(M)=\lambda_{2^*}.

By using a direct minimizing procedure, it can be shown that for 2 < s < 2^*, there exists a smooth positive function u_s such that its L^s-norm is equal to one, Q_s(u_s) = \lambda_s, and u_s satisfies the equation

\displaystyle \Delta_gu_s-\frac{n-2}{4(n-1)}R_gu_s+\lambda_su^{s-1}_s=0,\quad \text{in}\;M.

The direct method does not work when s=2^* because the Sobolev embedding H^1(M) \hookrightarrow L^{2^*}(M)  is continuous but not compact. However, if one can show that u_s is uniformly bounded, i.e. there exists a positive constant c such that u_s \le c in M for 2 < s < 2^*, then there exists a sequence \{s_i\} \subset (2, 2^*) such that and u_{s_i} converges to a smooth positive function u which satisfies the Yamabe equation .

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June 28, 2011

The Yamabe problem: A Story

Filed under: PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 15:47

I want to write a short survey about the Yamabe problem. Long time ago, I introduced the problem in this blog [here] but it turns out that the note was not rich enough to perform the importance of the problem.

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

Conjecture. Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere

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

Theorem (Yamabe). On a compact Riemannian manifold (M, g) of dimension \geqslant 3, there exists a metric g' conformal to g, such that the corresponding scalar curvature \text{Scal}_{g'} is constant.

As can be seen, the Yamabe problem is a special case of the prescribing scalar curvature problem that can be completely solved. For the prescribing scalar curvature, we also solve it completely when the invariant is non-positive.

1. Conformal metrics.

Definition (conformal). Two pseudo-Riemannian metrics g and \widetilde g on a manifold M are said to be

  • (pointwise) conformal if there exists a C^\infty function f on M such that

    \displaystyle \widetilde g=e^{2f}g;

  • conformally equivalent if there exists a diffeomorphism \alpha of M such that \alpha^* \widetilde g and g are pointwise conformal.

Note that, if g and \widetilde g are conformally equivalent, then \alpha is an isometry from e^{2f}g onto \widetilde g. So we will only study below the case \widetilde g = e^{2f}g.

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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

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