Dyson Brownian Motion and motion by mean curvature

We construct Dyson Brownian motion for $\beta \in (0,\mathrm{\infty }]$ by adapting the extrinsic construction of Brownian motion on Riemannian manifolds to the geometry of group orbits within the space of Hermitian matrices. When $\beta$ is infinite, the eigenvalues evolve by Coulombic repulsion and the group orbits evolve by motion by (minus one half times) mean curvature.