Ngô Quốc Anh

November 6, 2013

The prescribed scalar curvature: An inf x sup inequality in the positive Yamabe invariant

Filed under: PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 0:56

Today, we talk about an inequality of the form \inf \times \sup \geqslant c for solutions of the so-called prescribed scalar curvature problem, i.e.

\displaystyle -\frac{4(n-1)}{n-2}\Delta_g u + \text{Scal}_g u = V u^{2^\star -1}

where the scalar curvature \text{Scal}_g>0 is fixed but the function V which satisfies

0<a \leqslant V(x) \leqslant b

for any x \in M. Here 2^\star = 2n/(n-2) is the Sobolev critical exponent and n \geqslant 3 is the dimension of M. In addition, the manifold M is compact and has no boundary.

Theorem (Bahoura). There exists a positive constant c depending on a,b,M such that

\displaystyle \sup_M u \times \inf_M u \geqslant c

for any solution u of the PDE.

We prove this theorem using contradiction: There exists a sequence of solutions u_i of

\displaystyle -\frac{4(n-1)}{n-2}\Delta_g u_i + \text{Scal}_g u_i = V_i u_i^{2^\star -1}

such that

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November 3, 2013

The Yamabe problem: The case of spheres

Filed under: PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 23:54

When the manifold (M,g) is the standard sphere (\mathbb S^n, g_{\rm car}), the Yamabe is interesting. First, we cover the so-called Kazdan-Warner obstruction. The original statement is for the prescribing scalar curvature problem, i.e. the following PDE

\displaystyle -\frac{4(n-1)}{n-2}\Delta_g \phi + R\phi = R' \phi^{(n+2)/(n-2)}

where R and R' are scalar curvatures of the metrics g and g'=\phi^{4/(n-2)}g respectively. We note that, instead of using the condition \mu_{2n/(n-2)} = n(n-1)\omega_n^{2/n} to avoid the triviality of the limiting solution, in this generalized equation, one has to use

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

Theorem (Kazdan-Warner obstruction). If \phi >0 is a solution of the preceding PDE on (\mathbb S^n, g_{\rm car}) with n\geqslant 0, then

\displaystyle \int_{\mathbb S^n} \phi^{2n/(n-2)}\langle \nabla^i R' , \nabla_i F \rangle dv_g = 0

for any spherical harmonics F of degree 1.

When n=2, such an obstruction has been mentioned once here. We shall not provide any proof for this obstruction here, it is similar to that for the case n=2 and basically is integration by parts. We note that all spherical harmonics F of degree 1 satisfy

\displaystyle \nabla_{ij} F = -\frac{\lambda_1 F}{n} g_{ij} =: -\alpha^2 Fg_{ij}.

Interestingly, on the sphere (\mathbb S^n, g) having unit volume, constants appearing in the Sobolev inequality are optimal

\displaystyle \|\phi\|_{2n/(n-2)}^2 \leqslant K(n,2)^2 \|\nabla\phi\|_2^2 + \|\phi\|_2^2.

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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

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November 1, 2013

The Yamabe problem: The work by Thierry Aubin

Filed under: PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 22:05

Following the previous note about the work of Trudinger, today we talk about the work of Aubin regarding to the Yamabe problem, that is the following simple PDE

\displaystyle -\Delta \varphi + R\varphi = C_0 \varphi^\frac{n+2}{n-2}.

In his elegant paper entitlde “Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire” published in J. Math. Pures Appl. in 1976, Aubin proved the existence for almost all manifolds for n\geqslant 6.

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),

Aubin proved that

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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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May 16, 2011

Some properties of the Yamabe equation in the null case

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 19:56

Let us consider the Yamabe equation in the null case, that is

\displaystyle -\Delta u = f u^\frac{n+2}{n-2}, \quad x \in M

where M is a compact manifold of dimension n without boundary. We assume that u>0 is a smooth positive solution.

Since the manifold is compact without the boundary, the most simple result is

\displaystyle\int_M f u^\frac{n+2}{n-2}=0

by integrating both sides of the equation. Now we prove that

\displaystyle\int_M f <0.

Indeed, multiplying both sides of the PDE with u^{-\frac{n+2}{n-2}} and integrating over M, one obtains

\displaystyle -\int_M(\Delta u) u^{-\frac{n+2}{n-2}} = \int_M f .

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May 11, 2011

On the uniqueness for the Yamabe problem

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 14:25

The Yamabe problem has been discussed here. Basically, starting from a metric g, for a given constant R Yamabe wanted to show there always exists a positive function \varphi such that the scalar curvature of metric \overline g defined to be \varphi^\frac{4}{n-2}g equals R. In terms of PDEs, the scalar curvature satisfies the equation (called Yamabe equation)

\displaystyle \frac{{4\left( {n - 1} \right)}} {{n - 2}}\Delta \varphi + R\varphi = \overline R {\varphi ^{\frac{{n + 2}} {{n - 2}}}}

where \overline R the scalar curvature of metric \overline g. It is not hard to see that in the negative and null cases, two solutions of the Yamabe equation with \overline R = {\rm const} are proportional. Let us prove the following

Theorem. In the negative and null cases, two solutions of the Yamabe equation with R=\overline R = {\rm const} are constant 1.

We first fix a background metric g_0 and let \varphi_0 be a solution to the Yamabe equation with \overline R=\mu. That is, the scalar curvature of metric \varphi_0^\frac{4}{n-2}g_0 equals \mu.

We now consider a new metric g_1 still sitting in the same conformal class [g_0] of g_0 with conformal factor \varphi_1 such that its scalar curvature equals \mu. In other words, \varphi_1 is also a solution of the Yamabe equation with R=\overline R=\mu. Keep in mind

\displaystyle \frac{{4\left( {n - 1} \right)}} {{n - 2}}\Delta \varphi_1 + \mu\varphi_1 = \mu_1 {\varphi_1 ^{\frac{{n + 2}} {{n - 2}}}}.

We have two cases.

  • Suppose \mu=0. From the equation above, \Delta \varphi_1=0, that is, \varphi_1 is constant.
  • Suppose \mu<0. At a point P where \varphi_1 is maximum, \Delta \varphi_1 \geqslant 0, thus \varphi_1(P) \leqslant 1. Similarly, at a point Q where \varphi_1 is minimum, \Delta \varphi_1 \leqslant 0, thus \varphi_1(Q) \geqslant 1. Consequently, \varphi_1 \equiv 1.

Notice that the uniqueness result in the positive  case is no longer true in general. Due to Obata, this is true for Einstein manifolds, that is, when the Ricci curvature and metric are proportional.

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