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Structure of branch sets of harmonic functions and minimal submanifolds

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I will discuss some recent results on the structure of the branch set of multiple-valued solutions to the Laplace equation and minimal surface system. It is known that the branch set of a multiple-valued solution on a domain in $\mathbb{R}^n$ has Hausdorff dimension at most $n-2$. We investigate the fine structure of the branch set, showing that the branch set is countably $(n-2)$-rectifiable. Our result follows from the asymptotic behavior of solutions near branch points, which we establish using a modification of the frequency function monotonicity formula due to F. J. Almgren and an adaptation to higher-multiplicity of a “blow-up” method due to L. Simon that was originally applied to “multiplicity one” classes of minimal submanifolds satisfying an integrability hypothesis.

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