Ngô Quốc Anh

April 1, 2013

A proof of the uniqueness of solutions of the Lichnerowicz equations in the compact case

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 4:38

In this small note, we aim to derive some uniqueness property of solutions of the following PDE

$\displaystyle -a\Delta_g u +\text{Scal}_g u =|\sigma|_g^2 u ^{-2\kappa-3}-b\tau^2 u^{2\kappa+1}$

on a compact manifold $(M,g)$ where $\kappa=\frac{2}{n-2}$, $a=2\kappa+4$, and $b=\frac{n-1}{n}$.

We assume that $\phi_1$ and $\phi_2$ are solutions of the above PDE. Setting $\phi=\frac{\phi_2}{\phi_1}$. We wish to prove that $\phi=1$.

Let us consider the following trick basically due to David Maxwell, see this paper. Let $\widetilde g= \phi_1^{2\kappa}g$. Then the well-known formula for the Laplace-Beltrami operator $\Delta_g$, which is

$\displaystyle {\Delta _g}u = \frac{1}{{\sqrt {\det g} }}{\partial _i}(\sqrt {\det g} {g^{ij}}{\partial _j}u)$

helps us to write

$\displaystyle {\Delta _{\widetilde g}}u = \phi _1^{ - 2\kappa } \Big({\Delta _g}u + \frac{2}{{{\phi _1}}}{\nabla _g}{\phi _1}{\nabla _g}u \Big).$

Consequently,

$\displaystyle {\Delta _g}\phi = \phi _1^{2\kappa }{\Delta _{\widetilde g}}\phi - \frac{2}{{{\phi _1}}}{\nabla _g}{\phi _1}{\nabla _g}\phi .$

We now calculate $- a{\Delta _g}{\phi _2} + {R_g}{\phi _2}$ as follows.