Approximately j-dimensional Koch type sets are potentially minimal surfaces

2006/08/21 by Amos N. Koeller, Koeller, Amos N.
Mathematics · #28A78 #28A80 #49Q20 #53A10 #Classical Analysis and ODEs (math.CA) #Differential Geometry (math.DG) #FOS: Mathematics #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.math/0608528

openalex publication_date 2006/08/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We investigate the approximate j-dimensionality of the singularity sets of minimal surfaces prescribed by Simon. This leads to the clasification of 8 variations of approximately j-dimensional surfacs in terms of dimension and locally finite Hausdorff measure. We show that the singularity sets must either be well behaved or essentially purely unrectifiable. Examples of the unrectifiable sets that could occur are constructed and generalised. They are shown to be pseudo-fractal and related to the well known Koch Sets. Various representations of the measure and dimension of the sets are shown, as well as a fine balance in the spiralling nature of the sets.

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