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 $\xi \in H^1(M)$.

April 11, 2012

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

Filed under: Luận Văn, PDEs, Riemannian geometry — Tags: , — 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}} }}

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.