Geometric Analysis and Topology Seminar
Area minimizing hypersurfaces mod (p): A geometric free boundary problem
Speaker: Jonas Hirsch, Universität Leipzig
Location: Online
Videoconference link: https://nyu.zoom.us/s/99465062960
Date: Wednesday, November 17, 2021, 11 a.m.
Synopsis:
In this talk I would like to give an idea of our resent result on the structure of area minimizing hypersurfaces mod (p).
Motivation: If one considers real soap films one notice that from time to time one can find configurations where different soap films join on a common piece. One possibility to allow this kind of phenomenon is to consider flat chains with coefficients in Z/p. For instance for p = 2 one can deal with unoriented surfaces, for p = 3 one allows triple junctions. Using known results it can be shown that for p = 3 this common piece is itself nicely regular. It was our aim to investigate the situation for higher p.
We consider area minimizing m-dimensional currents mod (p) in complete C2 Riemannian manifolds Σ of dimension m + 1. For odd moduli we prove that, away from a closed rectifiable set of codimension 2, the current in question is, locally, the union of finitely many smooth minimal hypersurfaces coming together at a common C1,α boundary of dimension m − 1, and the result is optimal. For even p such structure holds in a neighborhood of any point where at least one tangent cone has (m − 1)-dimensional spine. These structural results are indeed the byproduct of a theorem that proves (for any modulus) uniqueness and decay towards such tangent cones. The underlying strategy of the proof is inspired by the techniques developed by Simon in a class of multiplicity one stationary varifolds. The major difficulty in our setting is produced by the fact that the cones and surfaces under investigation have arbitrary multiplicities ranging from 1 to ⌊p/2⌋.
We want to mention that the presented result can as well derived by the theory for stable varifolds developed by N. Wickramasekera.