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Structure of branch sets of harmonic functions and minimal submanifolds

  • Brian Krummel (University of Cambridge, United Kingdom)
A3 01 (Sophus-Lie room)

Abstract

I will discuss the fine structure of the branch set of multiple-valued solutions to the Laplace equation and minimal surface system. It was previously known that the branch set of a multiple-valued solution on a domain in $\mathbb{R}^n$ has Hausdorff dimension at most $n-2$. In joint work with Neshan Wickramasekera, we show that the branchset is countably $(n-2)$-rectifiable. This 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. In ongoing independent work, I show that the branch set decomposes into locally real analytic submanifolds along which the frequency is constant and points where the frequency is not lower semicontinuous.

Katja Heid

MPI for Mathematics in the Sciences Contact via Mail

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