In the early 1980's Almgren developed a theory of Dirichlet energy minimizing multi-valued functions, proving that the Hausdorff dimension of the singular set (including branch points) of such a function is at most $(n-2),$ where $n$ is the dimension of its domain. Almgren used this result in an essential way to show that the same upper bound holds for the dimension of the singular set of an area minimizing $n$-dimensional rectifiable current of arbitrary codimension. In either case, the dimension bound is sharp. We develop estimates to study the asymptotic behaviour of a multi-valued Dirichlet energy minimizer on approach to its singular set. Our estimates imply that a Dirichlet energy minimizer at ${\mathcal H}^{n-2}$ a.e. point of its singular set has a unique set of homogeneous multi-valued cylindrical tangent functions (blow-ups) to which the minimizer, modulo a set of single-valued harmonic functions, decays exponentially fast upon rescaling. A corollary is that the singular set 查看全文>>