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What is the exact meaning of "generic" in Colding-Minicozzi's paper "Generic mean curvature flow I; generic singularities"? (DOI link, arXiv link) Is there a specific explanation somewhere? I did not find it in that paper. Does someone know a paper or a book having a expanation of it? Thanks.

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  • $\begingroup$ It appears they are using it in the sense of general position. i.e. a feature of a metric is "generic" if it also holds for all nearby metrics. $\endgroup$ Apr 19 at 18:47

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I think most genericity in differential geometry is in the Baire category sense, i.e., a set is generic if it contains the intersection of a countable number of open dense subsets.

Thus the question is what topology they are using here. For the space of hypersurfaces, it usually means the $C^\infty$-topology, that is, locally they could be written as graphs and they converge in any $C^k$-norm.

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