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arXiv · 2610.35755

The size of an overtwisted disk

Álvaro del Pino Gómez·Lukas Nakamura
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Abstract

We introduce a notion of size for an overtwisted disk, defined in terms of the Shelukhin-Hofer energy required to stabilise Legendrians in its vicinity. We compute this size for the standard $3$-dimensional overtwisted disk, which we use to prove that certain explicit product hyperdisks in higher-dimensional contact manifolds are overtwisted (and their overtwisted size can be computed explicitly as well). We then use the overtwisted size to establish a number of quantitative results on contact flexibility. Concretely, we (1) obtain quantitative $h$--principles for Legendrians in the complement of an overtwisted disk, (2) construct overtwisted-at-infinity contact manifolds whose Legendrian Shelukhin-Hofer metric vanishes, (3) estimate overtwisted size under contact products and under products with Liouville manifolds, and (4) derive action bounds for contractible Reeb orbits in closed overtwisted contact manifolds.

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Álvaro del Pino Gómez, Lukas Nakamura. 2026-10-26. The size of an overtwisted disk. https://arxiv.org/abs/2610.35755

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