Riemannian geometry « Ngô Quốc Anh

April 16, 2012

The Yamabe problem: The work by Neil Sidney Trudinger

Filed under: PDEs, Riemannian geometry — Tags: Yamabe problem — 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 $\xi \in H^1(M)$ .

(more…)

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April 11, 2012

Existence results for the Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds

Filed under: Luận Văn, PDEs, Riemannian geometry — Tags: Lichnerowicz equations — Ngô Quốc Anh @ 2:18

A couple of days ago, I got an acceptance for publication in Advances in Mathematics journal that makes me feel so exciting because of the prestige of the journal. This is part of my PhD thesis in NUS under the supervision of professor Xu. Besides, this is joint work with him.

The work looks like simple, I mean, we just try to solve the following PDE

$\displaystyle {\Delta _g}u + hu = f{u^{{2^\star} - 1}} + \frac{a}{{{u^{{2^\star} +1}}}}, \quad u>0,$

where $\Delta_g=-{\rm div}_g(\nabla_g)$ is the Laplace-Beltrami operator, $2^\star=\frac{2n}{n-2}$ is the critical Sobolev exponent, $M$ is a compact manifold without boundary of dimension $n \geqslant 3$ , and $h$ , $f$ , $a \geqslant 0$ are smooth functions. In our work, the above PDE is numbered as (1.2). I don’t want to mention the physical background of the equation, in a few words, this equation is motivated by the Hamiltonian constraint equations of General Relativity through the so-called conformal method. Apparently, the important and frequently studied prescribing scalar curvature equation is just a particular case.

In this work, we focus on the negative Yamabe-scalar field invariant, that is, $h<0$ . Our result basically consists of two theorems.

In the first result, we consider the case that $f$ may change its sign, we prove

Theorem 1.1. Let $(M,g)$ be a smooth compact Riemannian manifold without the boundary of dimension $n \geqslant 3$ . Assume that $f$ and $a \geqslant 0$ are smooth functions on $M$ such that $\int_M f dv_g<0$ , $\sup f > 0$ , $\int_M a dv_g >0$ , and $|h| < \lambda _f$ where $\lambda_f$ is given in (2.1) below. Let us also suppose that the integral of $a$ satisfies

$\displaystyle\int_M {ad{v_g}} < \frac{1}{n-2}{\left( {\frac{{n - 1}}{n-2}} \right)^{n - 1}}{\left( {\frac{{|h|}}{{\int_M {|{f^ - }|d{v_g}} }}} \right)^n}\int_M {|{f^ - }|d{v_g}}$

where $f^-$ is the negative part of $f$ . Then there exists a number $C > 0$ to be specified such that if

$\displaystyle\frac{{\sup {f }}}{{\int_M {{|f^ -| }d{v_g}} }} <C,$

Equation (1.2) possesses at least two smooth positive solutions.

In the next result, we consider the case that $f \leqslant 0$ . In this case, we are able to get a complete characterization of the existence of solutions. More precisely, we prove

Theorem 1.2. Let $(M,g)$ be a smooth compact Riemannian manifold without boundary of dimension $n \geqslant 3$ . Let $h<0$ be a constant, $f$ and $a$ be smooth functions on $M$ with $a \geqslant 0$ in $M$ , $f \leqslant 0$ but not strictly negative. Then Equation (1.2) possesses one positive solution if and only if $|h|<\lambda_f$ .

As one can see, the above theorem does not allow $f$ to be strictly negative. Fortunately, our approach can cover this case too. This is the last remark in the paper as we prove the following: if $f<0$ then Equation (1.2) always possesses one positive solution, I mean, without any condition on $f$ except the condition $f<0$ .

It is important to note that in the case $f \leqslant 0$ , the solution is always unique by using the monotone trick.

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March 20, 2012

The Yamabe problem: The work by Hidehiko Yamabe

Filed under: PDEs, Riemannian geometry — Tags: Yamabe problem — 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

$\displaystyle -\Delta \varphi_q + R\varphi_q = \mu_q \varphi_q^\frac{n+2}{n-2}.$

(more…)

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November 16, 2011

Conformal compactification

Filed under: Riemannian geometry — Ngô Quốc Anh @ 0:49

Start with a pseudo-Riemannian manifold $(M,g)$ , let $\tilde{g}$ be another pseudo-Riemannian metric on $M$ , we say that $g$ and $\tilde{g}$ are conformal if there exists a positive scalar function $\phi$ on $M$ such that $\tilde{g} = \phi g$ (sufficient smoothness of the relevant quantities are always assumed).

Observe that two conformal metrics measure angles the same way: recall that on a pseudo-Riemannian manifold $(M,g)$ , given a point $p\in M$ and two non-null vectors $v,w\in T_pM$ , the angle between the vectors can be defined by

$\displaystyle \frac{g(v,w)^2}{g(v,v) g(w,w) }.$

(Notice that on Euclidean space, if $v,w$ form an angle $\theta$ , then $v\cdot w = |v||w| \cos\theta$ .) Thus if $\tilde{g}$ is conformal to $g$ , they define the same angles

$\displaystyle \frac{\tilde{g}(v,w)^2}{\tilde{g}(v,v)\tilde{g}(w,w)} = \frac{\phi^2 g(v,w)^2}{\phi g(v,v) \phi g(w,w)} = \frac{g(v,w)^2}{g(v,v)g(w,w)}$

In fact, this inference works the other way too. If $g$ , $\tilde{g}$ are two pseudo-Riemannian metrics such that for any two vectors $v,w$ we have $g(v,w) = 0 \iff \tilde{g}(v,w) = 0$ , then $g$ , $\tilde{g}$ are conformal (up to a change of sign) by the above definition (see e.g. Exercise 14, Chapter 2 from B.O’Neill, Semi-Riemannian Geometry).

So, in plain English, two metrics are conformal if they measure angles the same way.

Now, let $(M,g)$ be a pseudo-Riemannian manifold that is non-compact. A conformal compactification of $(M,g)$ is a choice of a metric $\tilde{g}$ such that $(M,\tilde{g})$ can be isometrically embedded into a compact domain $\tilde{M}$ of a pseudo-Riemannian manifold $(M',g')$ (well, I am ignoring some regularity issues here). Let $\phi$ be the conformal factor as before. Then observe that any regular extension of $\phi$ to the conformal boundary $\partial\tilde{M} \subset M'$ must vanish on said boundary. This reflects the property of a conformal compactification that “brings infinity to a finite distance”.

The simplest example of conformal compactification is the one-point compactification of Euclidean space via the stereographic projection. In this case, the target manifold $(M',g')$ is compact itself, taken to be standard sphere. The source manifold $(M,g)$ is Euclidean space with the standard metric, and the image set $\tilde{M}$ is taken to be the sphere minus the north pole.

[Source]

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November 8, 2011

A blowup proof of the Aubin theorem in the Yamabe problem

Filed under: PDEs, Riemannian geometry — Tags: Yamabe problem — 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 .

(more…)

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November 1, 2011

An ODE appearing in the Nirenberg problem

Filed under: PDEs, Riemannian geometry — Ngô Quốc Anh @ 20:21

It is well-known that the simplest form of the Nirenberg problem is equivalent to solving the following PDE

$-\Delta u + 2= e^u$

in $\mathbb S^2$ . Using stereographic projection, one can see that the above PDE is equivalent to

$-\Delta u = e^u$

in $\mathbb R^2$ . If we assume that the solution $u$ has finite energy in the sense that

$\displaystyle \int_{\mathbb R^2} u <+\infty,$

it is well-known that the preceding PDE has unique radial solution. In terms of ODE language, our PDE can be rewritten as

$\displaystyle -u''(r)-\frac{1}{r}u'(r)=e^{u(r)},\quad r\geqslant 0.$

The purpose of this note is to find solutions to the above ODE. Our approach consists of several steps as shown below.

Step 1. Let $r = e^z$ . We then have $u(r)=u(e^z)=v(z)$ which implies that

$\displaystyle u'(r) = e^{-z}v'(z), \quad u''(r) = e^{-2z}(v''(z)-v'(z)).$

(more…)

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October 8, 2011

Locally conformally flat manifolds and Weyl and Cotton tensors, 2

Filed under: Riemannian geometry — Ngô Quốc Anh @ 3:27

The purpose of this note is to prove the following result that left in the previous entry

Lemma. Provided the Weyl tensor vanishes, equation

$\displaystyle {\nabla _i}{\nabla _j}f - {\nabla _i}f{\nabla _j}f + \frac{1}{2}|\nabla f{|^2}{g_{ij}} = {S_{ij}}$

is locally solvable if and only if the following integrability condition is satised

$\displaystyle {\nabla _k}{S_{ij}} = {\nabla _i}{S_{kj}}.$

That is, if and only if the Cotton tensor vanishes.

Proof. It is necessary and suffcient to find a 1-form $X$ locally such that

$\displaystyle {\nabla _i}{X_j} = {c_{ij}} = {S_{ij}} + {X_i}{X_j} - \frac{1}{2}|X{|^2}{g_{ij}},$

where $c = c (X, g)$ is a symmetric 2-tensor depending only on $X$ and $g$ . To see this, by the symmetry of the RHS, we have

$\displaystyle \nabla _iX_j=\nabla _jX_i$

(more…)

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October 4, 2011

Locally conformally flat manifolds and Weyl and Cotton tensors

Filed under: Riemannian geometry — Ngô Quốc Anh @ 20:45

The purpose of this note is to prove the following

Theorem. A Riemannian manifold $(M^n, g)$ is locally conformally flat if and only if

for $n \geqslant 4$ , the Weyl tensor vanishes;

for $n=3$ , the Cotton tensor vanishes.

To this purpose, let us briefly recall some definitions

The Weyl tensor. The Weyl tensor can be defined using the following formula

$\displaystyle W = \text{Rm} - \frac{\text{Scal}}{{2(n - 1)n}}g \odot g - \frac{1}{{n - 2}}\left( {\text{Ric} - \frac{\text{Scal}}{n}g} \right) \odot g$

where $n\geqslant 3$ and $\odot$ denotes the Kulkarni–Nomizu product of two symmetric (0,2) tensors. Writing the Weyl tensor in this way means that the Weyl tensor is actually a (0,4) tensor. It can be seen that the Weyl tensor can be rewritten in this form

$\displaystyle W = \text{Rm} - \frac{1}{{n - 2}}\left( {\text{Ric} - \frac{g}{{2(n - 2)}}\text{Scal}} \right) \odot g$

where the part

$\displaystyle S = \frac{1}{{n - 2}}\left( {{\text{Ric}} - \frac{g}{{2(n - 2)}}{\text{Scal}}} \right) \odot g$

is called the Schouten tensor. We have the following result

(more…)

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September 5, 2011

The Riemannian Penrose inequality

Filed under: Riemannian geometry — Ngô Quốc Anh @ 1:43

In mathematical general relativity, the Penrose inequality, first conjectured by Sir Roger Penrose, estimates the mass of a spacetime in terms of the total area of its black holes and is a generalization of the positive mass theorem. The Riemannian Penrose inequality is the most important special case. Specifically, if $(M, g)$ is an asymptotically flat Riemannian 3-manifold with nonnegative scalar curvature and ADM mass $m$ , and $A$ is the area of the outermost minimal surface (possibly with multiple connected components), then the Riemannian Penrose inequality asserts

$\displaystyle m \geq \sqrt{\frac{A}{16\pi}}.$

This is purely a geometrical fact, and it corresponds to the case of a complete three-dimensional, space-like, totally geodesic submanifold of a (3 + 1)-dimensional spacetime. Such a submanifold is often called a time-symmetric initial data set for a spacetime. The condition of $(M, g)$ having nonnegative scalar curvature is equivalent to the spacetime obeying the dominant energy condition.

This inequality was first proved by Gerhard Huisken and Tom Ilmanen in 1997 [here and here] in the case where $A$ is the area of the largest component of the outermost minimal surface. Their proof relied on the machinery of weakly defined inverse mean curvature flow, which they developed. In 1999, Hubert Bray [here] gave the first complete proof of the above inequality using a conformal flow of metrics. Both of the papers were published in 2001 in the Journal of Differential Geometry.

Source: Wiki

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August 9, 2011

Stereographic projection, 5

Filed under: PDEs, Riemannian geometry — Ngô Quốc Anh @ 14:39

I put here several common formulas that are needed when we work on stereographic projection. First we try to calculate

$\displaystyle\Delta \left( {{{\left( {\frac{2}{{1 + |x{|^2}}}} \right)}^{n - 2}}} \right).$

We do step by step.

Step 1. A direct computation leads us to

$\displaystyle\begin{gathered} {\partial _i}\left( {{{\left( {\frac{2}{{1 + |x{|^2}}}} \right)}^{n - 2}}} \right) = (n - 2){\left( {\frac{2}{{1 + |x{|^2}}}} \right)^{n - 3}}{\partial _i}\left( {\frac{2}{{1 + |x{|^2}}}} \right) \hfill \\ \qquad= (n - 2){\left( {\frac{2}{{1 + |x{|^2}}}} \right)^{n - 3}}\frac{{ - 4{x_i}}}{{{{(1 + |x{|^2})}^2}}}. \hfill \\ \end{gathered}$

Therefore,

(more…)

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Ngô Quốc Anh

April 16, 2012

The Yamabe problem: The work by Neil Sidney Trudinger

April 11, 2012

Existence results for the Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds

March 20, 2012

The Yamabe problem: The work by Hidehiko Yamabe

November 16, 2011

Conformal compactification

November 8, 2011

A blowup proof of the Aubin theorem in the Yamabe problem

November 1, 2011

An ODE appearing in the Nirenberg problem

October 8, 2011

Locally conformally flat manifolds and Weyl and Cotton tensors, 2

October 4, 2011

Locally conformally flat manifolds and Weyl and Cotton tensors

September 5, 2011

The Riemannian Penrose inequality

August 9, 2011

Stereographic projection, 5

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