Approximately j-dimensional Koch type sets are potentially minimal surfaces

Details Approximately j-dimensional Koch type sets are potentially minimal surfaces Approximately j-dimensional Koch type sets are potentially minimal surfaces

2006-08-21

arxiv.org/abs/math/0608528v1

Mathematics

Classical Analysis and ODEs

155 pages, Section 1 of the PhD thesis of the author

Scientific paper

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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