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We are interested in the stability of a class of totally geodesic wave maps,as recently studied by Abbrescia and Chen, and later by Duan and Ma. Therelevant equations of motion are a system of coupled semilinear wave andKlein-Gordon equations in $\mathbb{R}^{1+n}$ whose nonlinearities are criticalwhen $n=2$. In this paper we use a pure energy method to show global existencewhen $n=2$. By carefully examining the structure of the nonlinear terms, we areable to obtain uniform energy bounds at lower orders. This allows us to provepointwise decay estimates and also to reduce the required regularity.

Stability of some two dimensional wave maps: a wave--Klein-Gordon model