Instantaneous Hamiltonian displaceability and arbitrary symplectic squeezability for critically negligible sets (2408.17444v2)

Abstract: We call a metric space $s$-negligible iff its $s$-dimensional Hausdorff measure vanishes. We show that every countably $m$-rectifiable subset of $\mathbb{R}^{2n}$ can be displaced from every $(2n-m)$-negligible subset by a Hamiltonian diffeomorphism that is arbitrarily $C^\infty$-close to the identity. As a consequence, every countably $n$-rectifiable and $n$-negligible subset of $\mathbb{R}^{2n}$ is arbitrarily symplectically squeezable. Both results are sharp w.r.t. the parameter $s$ in the $s$-negligibility assumption. The proof of our squeezing result uses folding. Potentially, our folding method can be modified to show that the Gromov width of $B^{2n}_1\setminus A$ equals $\pi$ for every countably $(n-1)$-rectifiable closed subset $A$ of the open unit ball $B^{2n}_1$. This means that $A$ is not a barrier.

• The paper introduces key results on instantaneous Hamiltonian displaceability and arbitrary symplectic squeezability for specific subsets in symplectic geometry.
• The authors demonstrate that small sets meeting certain rectifiability and negligibility conditions have zero displacement energy, enabling instantaneous Hamiltonian displacement.
• They also show that certain critically negligible sets can be embedded into any other symplectic manifold of the same dimension, revealing significant symplectic flexibility.

Insight into Instantaneous Hamiltonian Displaceability and Symplectic Squeezability

The paper authored by Yann Guggisberg and Fabian Ziltener explores symplectic geometry, exploring the properties of subsets of Euclidean space under symplectic transformations, specifically displacement and squeezing phenomena. It extends fundamental understandings in symplectic topology in regards to the dimension and geometric properties of subsets, providing new insights into the behavior of these subsets under symplectic transformations.

The paper introduces two main results: The first is on the instantaneous Hamiltonian displaceability of small subsets, and the second concerning the arbitrary symplectic squeezability of certain critically negligible sets.

Instantaneous Hamiltonian Displaceability

The authors focus on subsets of $\mathbb{R}^{2n}$ that are countably $m$-rectifiable and $(2n-m)$-Hausdorff negligible. They establish that such sets can be instantaneously displaced from each other using Hamiltonian dynamics. This result is significant as it provides a precise condition under which small sets can be symplectically disjoint even if their configurations or dimensions suggest otherwise.

Crucially, the authors prove that the displacement energy for subsets with dimension $\leq n$ is zero when these subsets meet certain rectifiability and negligibility conditions. They investigate scenarios where sets are cannot physically overlap due to their geometric and symplectic properties. This contributes to the broader discourse on the size and properties of sets that do not admit such geometrical embedding (or constraints).

Arbitrary Symplectic Squeezability

The second result pertains to symplectic squeezability in critically negligible sets. Specifically, Guggisberg and Ziltener show that every countably $n$-rectifiable and $n$-negligible bounded subset of $\mathbb{R}^{2n}$ can be embedded into any other symplectic manifold of the same dimension. This discovery implies a new level of flexibility as opposed to rigidity in symplectic embeddings, reaffirming the idea that symplectic geometry is exceedingly flexible below half the dimensional threshold.

Implications and Future Directions

The results have profound implications, potentially altering the approach to symplectic and Hamiltonian dynamics in geometric analysis. The concept of displaceability offers new intuition for object recognition within symplectic categories. Furthermore, the methodology of employing Hamiltonian diffeomorphisms and folding mechanisms provides powerful techniques for examining the broader inherent flexibilities of symplectic subsets.

Speculative Future Developments:

• AI and Computational Geometry: As AI continues to evolve, there is potential for machine learning algorithms to simulate and virtually manipulate symplectic embeddings, leveraging insights into instantaneous displacements and compressions to optimize and predict geometric transformations.
• Advanced Robotics: The flexibility of symplectic squeezes could inform robotic path planning and manipulation, especially in constrained environments where minute displacements are required.
• Quantum and Theoretical Physics: As symplectic geometry is foundational in formulating physical theories, deeper understanding in displacement energy and flexibility may refine existing models or inform novel quantum states.

In conclusion, the paper by Guggisberg and Ziltener makes a solid contribution to the theory of symplectic geometry, expanding the understanding of how small sets behave under symplectic transformations. Their work invites further investigation into symplectic flexibility, particularly in higher-dimensional setups, unraveling nuanced properties of geometric constructs in topology and related fields.