  We consider the motion by mean curvature of an $n$-dimensional graph over a
time-dependent domain in $\mathbb{R}^n$, intersecting $\mathbb{R}^n$ at a
constant angle. In the general case, we prove local existence for the
corresponding quasilinear parabolic equation with a free boundary, and derive a
continuation criterion based on the second fundamental form. If the initial
graph is concave, we show this is preserved, and that the solution exists only
for finite time. This corresponds to a symmetric version of triple junction
motion of hypersurfaces by mean curvature, with constant angles at the
junction.
