Regularity of minimal submanifolds and mean curvature flows with a common free boundary

Let $N$ be a smooth $(n+l)$-dimensional Riemannian manifold. We show that if $V$ is an area-stationary union of three or more $C^{1,\mu }$ $n$-dimensional submanifolds-with-boundary $M_k\subset N$ with a common boundary $\mathrm{\Gamma }$, then $\mathrm{\Gamma }$ is smooth and each $M_k$ is smooth up to $\mathrm{\Gamma }$ (real-analytic in the case $N$ is real-analytic). This extends a previous result of the author for codimension $l=1$. We additionally show that if $\{V_t{\}}_{t\in (-1,1)}$ is a Brakke flow such that each time-slice $V_t$ is a union of three or more $n$-dimensional submanifolds-with-boundary $M_{k,t}\subset N$ with a common boundary ${\mathrm{\Gamma }}_t$ and with parabolic $C^{2+\mu }$ regularity in time-space, then $\{{\mathrm{\Gamma }}_t{\}}_{t\in (-1,1)}$ and $\{M_{k,t}{\}}_{t\in (-1,1)}$ are smooth (second Gevrey with real-analytic time-slices in the case $N$ is real-analytic).