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We construct travelling wave graphs of the form $z=-ct+\phi(x)$, $\phi: x \in\mathbb{R}^{N-1} \mapsto \phi(x)\in \mathbb{R}$, $N \geq 2$, solutions to the$N$-dimensional forced mean curvature motion $V_n=-c_0+\kappa$ ($c\geq c_0$)with prescribed asymptotics. For any 1-homogeneous function $\phi_{\infty}$,viscosity solution to the eikonal equation$|D\phi_{\infty}|=\sqrt{(c/c_0)^2-1}$, we exhibit a smooth concave solution tothe forced mean curvature motion whose asymptotics is driven by$\phi_{\infty}$. We also describe $\phi_{\infty}$ in terms of a probabilitymeasure on $\mathbb{S}^{N-2}$.

Travelling graphs for the forced mean curvature motion in an arbitrary space dimension