Colding and Minicozzi have introduced a natural measure of the complexity of a submanifold of Euclidean space that they call its entropy. It can be shown that the entropy of a closed hypersurface is uniquely minimized by a round sphere and so it is natural to study the extent to which this rigidity property of the round spheres is stable. I will provide one perspective, given by Lu Wang and myself, that shows that closed surfaces in R^3 with entropy close to that of the round two-sphere are close as closed sets. Several generalizations and related questions will also be discussed.
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