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 <n. 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)}.

See also:

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