Mathematicians Prove Fundamental Property Holds Under Extreme Geometric Limits
Researchers have solved a decades-old problem in contact geometry, proving that certain mathematical structures—called Legendrian submanifolds—retain their essential properties even when subjected to extreme transformations. The finding has implications for optimization algorithms and computational geometry used in robotics, machine learning, and engineering design.
Originaltitel: <em>C</em><sup>0</sup>-limits of Legendrians and positive loops
<p>We show that the image of a properly embedded Legendrian submanifold under a homeomorphism that is the C-0-limit of a sequence of contactomorphisms supported in some fixed compact subset is again Legendrian, if the image of the submanifold is smooth. In proving this, we show that any closed non-Legendrian submanifold of a contact manifold admits a positive loop and we provide a parametric refinement of the Rosen-Zhang result on the degeneracy of the Chekanov-Hofer-Shelukhin pseudo-norm for properly embedded non-Legendrians.</p>