# Ngô Quốc Anh

## August 11, 2010

### Two ways to present SU(3) Toda systems

Filed under: PDEs, Riemannian geometry — Ngô Quốc Anh @ 4:13

This entry is concerned with SU(3) Toda systems in non-abelian relativistic self-dual gauge theory. Usually, they are given by $\displaystyle\begin{gathered} - {\Delta _g}{u_1} = 2{\lambda _1}\left( {\frac{{{e^{{u_1}}}}}{{\int_M {{e^{{u_1}}}d{v_g}} }} - \frac{1}{{|M|}}} \right) - {\lambda _2}\left( {\frac{{{e^{{u_2}}}}}{{\int_M {{e^{{u_2}}}d{v_g}} }} - \frac{1}{{|M|}}} \right), \hfill \\ - {\Delta _g}{u_2} = - {\lambda _1}\left( {\frac{{{e^{{u_1}}}}}{{\int_M {{e^{{u_1}}}d{v_g}} }} - \frac{1}{{|M|}}} \right) + 2{\lambda _2}\left( {\frac{{{e^{{u_2}}}}}{{\int_M {{e^{{u_2}}}d{v_g}} }} - \frac{1}{{|M|}}} \right), \hfill \\ \end{gathered}$

on $M$ with $\displaystyle\int_M {{u_1}d{v_g}} = \int_M {{u_2}d{v_g}} = 0$

where $(M,g)$ denotes a compact Riemannian surface and $\lambda_1$, $\lambda_2$ are non-negative constants.

We now consider function $h$ satisfying $\displaystyle - \Delta h =\frac{1}{|M|} \quad \text{ in } M$

The existence of such a $h$ will be considered later.

We now introduce $\displaystyle\begin{gathered} {v_1} = {u_1} - \log \int_M {{e^{{u_1}}}d{v_g}} - (2{\lambda _1} - {\lambda _2})h, \hfill \\ {v_2} = {u_2} - \log \int_M {{e^{{u_2}}}d{v_g}} - ( - {\lambda _1} + 2{\lambda _2})h. \hfill \\ \end{gathered}$