Absence of Self-Similar Blow-up and Local Well-posedness for the Constant Mean-Curvature Wave Equation

In this note, we consider the constant-mean-curvature wave equation in $(1+2)$-dimensions. We show that it does not admit any self-similar blow-up. We also remark that the equation is locally well-posed for initial data in $\dot{H}^{3/2}$.