Ngô Quốc Anh

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

Note that as we have already discussed in the previous note that the Sobolev inequality takes the following form $\displaystyle \|\varphi\|_{L^p} \leqslant \big(K(n,q)+\varepsilon \big) \|\nabla \varphi\|_{L^q} + A_q(\varepsilon) \|\varphi\|_q$

with $1/p=1/q-1/n>0$. When the sectional curvature is constant, $\varepsilon=0$ is allowed as the following holds $\displaystyle \|\varphi\|_{L^p} \leqslant K(n,q) \|\nabla \varphi\|_{L^q} + A_q(0) \|\varphi\|_q.$

In fact, Aubin obtained on $\mathbb S^n$ the following $\displaystyle \|\varphi\|_{L^p}^q \leqslant K(n,q)^q \|\nabla \varphi\|_{L^q}^q + A(q) \|\varphi\|_q^q$

if $1 \leqslant q \leqslant 2$ and $\displaystyle \|\varphi\|_{L^p}^{q/(q-1)} \leqslant K(n,q)^{q/(q-1)} \|\nabla \varphi\|_{L^q}^q + A(q) \|\varphi\|_q^{q/(q-1)}$

if $2 \leqslant q . Suppose $\mathbb S^n$ has unit volume, using the constant function $1$ we easily obtain $A(2) \geqslant 1$ and this answers why the constants are optimal.

It is important to note that these findings due to Aubin are not optimal at the moment. By a remarkable joint work with Vaugon, Hebey was able to prove that for any compact Riemannian manifold of dimension $n \geqslant 3$ $\displaystyle \|\phi\|_{2n/(n-2)}^2 \leqslant K(n,2)^2 \|\nabla\phi\|_2^2 + B\|\phi\|_2^2$

for some positive constant $B$.

For the sphere $(\mathbb S^n, g_{\rm car})$, there holds $\displaystyle \mu_{2n/(n-2)} = n(n-1)\omega_n^{2/n}.$

To see this, we can set $R'=1+\varepsilon \cos (\alpha r)$ where $\alpha^2=R/ \big( n(n-1) \big)$. Then on one hand, the equation $\displaystyle -\frac{4(n-1)}{n-2}\Delta_g \phi + R\phi = \big( 1+\varepsilon \cos (\alpha r) \big) \phi^{(n+2)/(n-2)}$

has no solution due to the Kazdan-Warner obstruction. However, on the other hand, $\displaystyle \mu_{2n/(n-2)} < n(n-1)\omega_n^{2/n} \big( 1+\varepsilon \cos (\alpha r) \big)^{-(n-2)/n}$

as long as $\displaystyle \mu_{2n/(n-2)} < n(n-1)\omega_n^{2/n}$ and $\varepsilon>0$ is small enough.

In the case $R'=R$, the equation $\displaystyle -\frac{4(n-1)}{n-2}\Delta_g \phi + R\phi = R \phi^{(n+2)/(n-2)}$

has infinitely many solutions. For example, $\phi (r)=\big( \beta - \cos (\alpha r) \big)^{1-n/2}$ with $\beta>1$ solves $\displaystyle -\frac{4(n-1)}{n-2}\Delta_g \phi + R\phi = (\beta^2-1) R \phi^{(n+2)/(n-2)}.$