Non-squeezing and capacities for some calibrated geometries
Kain Dineen, Spiro Karigiannis
TL;DR
This work extends Gromov’s non-squeezing and the associated capacity/rigidity framework from the classical symplectic setting to multisymplectic geometries given by $ω^k$ and to general calibrations, with a particular emphasis on the special Lagrangian case defined by the holomorphic volume form $Ω$. It provides two complementary proofs for the affine non-squeezing of $ω^k$ (one via a direct Barrow–Shafiee-style argument and another by reducing to the symplectic case through stabilizer analysis), and proves that the stabilizer of $ω^k$ is closed under transposition, yielding a reduction to the standard symplectic group in many cases. For calibrations, the paper identifies a sufficient (but not always necessary) condition—stabilizer closed under transposition—for affine non-squeezing to hold, and explores several calibrated geometries, including $G_2$ and special Lagrangian structures, highlighting both results and open questions. In the special Lagrangian setting, the authors establish affine non-squeezing for $Ω$ under complex-linear maps and derive rigidity statements, while signaling significant challenges in extending these results beyond the complex-linear regime or to higher phases. Overall, the work opens pathways to a broader capacity theory in calibrated geometry and invites further development of nonlinear analogues and stability questions across calibrations.
Abstract
It was shown by Barron--Shafiee that an analogue of Gromov's non-squeezing theorem holds for affine maps which preserve a power $ω^k$ of the symplectic form $ω$ on $\mathbb{R}^{2n}$. In the present paper, we state and prove in two ways an improved version of their result which is closer to the classical affine non-squeezing theorem. One proof closely follows their argument, and the other consists of a reduction to the classical case by showing that, except for the case $k = n$, every linear map that preserves $ω^k$ must be symplectic or anti-symplectic. We then study when a calibration form satisfies an (affine) non-squeezing theorem. Particular focus is given to the special Lagrangian case, where we are able to establish an affine non-squeezing theorem for the holomorphic volume form $Ω= dz^1 \wedge \cdots \wedge dz^n$. The classical symplectic affine rigidity theorem states roughly that a non-singular linear map is symplectic or anti-symplectic if and only if it preserves the "capacity" of every ellipsoid. We establish an affine special Lagrangian version of this theorem under the additional assumption that the map is complex-linear. We also discuss some natural future questions.