Structure of branch sets of harmonic functions and minimal submanifolds

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


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

Upcoming Events of This Seminar

  • Mar 12, 2024 tba with Theresa Simon
  • Mar 26, 2024 tba with Phan Thành Nam
  • Mar 26, 2024 tba with Dominik Schmid
  • May 7, 2024 tba with Manuel Gnann
  • May 14, 2024 tba with Barbara Verfürth
  • May 14, 2024 tba with Lisa Hartung
  • Jun 25, 2024 tba with Paul Dario
  • Jul 16, 2024 tba with Michael Loss