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.

(more…)

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