A. Sukhov, A. Tumanov
We introduce a method for constructing J-complex discs. The method only uses the standard scheme for solving the Beltrami equation and the Schauder principle. As an application, we give a short self-contained proof of Gromov's Non-Squeezing Theorem.
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We re-prove Gromov's non-squeezing theorem by applying Polyfold Theory to a simple Gromov-Witten moduli space. Thus we demonstrate how to utilize the work of Hofer-Wysocki-Zehnder to give proofs involving moduli spaces of pseudoholomorphic curves that are relatively short and broadly accessible, while also fully detailed and rigorous. We moreover review the polyfold description of Gromov-Witten moduli spaces in the relevant case of spheres with minimal energy and one marked point.
Locally conformally symplectic deformation of Gromov non-squeezing
本文将Gromov非压缩现象的形变理论推广至局部共形辛（lcs）结构，并猜想lcs几何中存在类似Eliashberg-Polterovich切触非压缩的类比。
This paper deals with the problem of generalising Gromov's non squeezing theorem to an infinite dimensional Hilbert phase space setting. By following the lines of the proof by Hofer and Zehnder of finite dimensional non-squeezing, we recover an infinite dimensional non-squeezing result by Kuksin for symplectic diffeomorphisms which are non-linear compact perturbations of a symplectic linear map. We also show that the infinite dimensional non-squeezing problem, in full generality, can be reformulated as the problem of finding a suitable Palais-Smale sequence for a distinguished Hamiltonian acti…
A remark on deformation of Gromov non-squeezing
本文研究了Gromov非挤压现象在辛形式微小变形下的持续性，通过特定方法证明了在某些情况下该现象仍然成立。
A proof of Gromov's non-squeezing theorem
本文给出了Gromov非挤压定理中伪全纯球模空间紧致性的两个证明，通过重新参数化曲线均匀分布辛面积并应用均值不等式或Gromov-Schwarz引数获得梯度界限，避免了气泡分析，使证明更初等。
We solve a problem on filling by Levi-flat hypersurfaces for a class of totally real 2-tori in a real 4-manifold with an almost complex structure tamed by an exact symplectic form. As an application we obtain a simple proof of Gromov's non-squeezing theorem in dimension 4 and new results on rigidity of symplectic structures.
We prove a generalization of Gromov's symplectic non-squeezing theorem for the case of Hilbert spaces. Our approach is based on filling almost complex Hilbert spaces by complex discs partially extending Gromov's results on existence of $J$-complex curves. We apply our result to the flow of the discrete nonlinear Schrödinger equation.
On certain quantifications of Gromov's non-squeezing theorem
本文研究穿孔 4 维球辛嵌入单位圆柱时缺失集合的大小，证明其 Minkowski 维数至少为 2，并通过构造及附录结果说明该下界的最优范围，同时讨论延拓嵌入下的最小体积。
After reformulating Gromov's non-squeezing theorem as an area-inequality, we discuss a seemingly natural higher dimensional generalization.
We find an upper bound for the Gromov width of coadjoint orbits of U(n) with respect to the Kirillov-Kostant-Souriau symplectic form by computing certain Gromov-Witten invariants. The approach presented here is closely related to the one used by Gromov in his celebrated non-squeezing theorem.
Gromov ground state in phase space engineering for fusion energy
该研究聚焦聚变能相空间工程的基本约束问题，指出格罗莫夫非挤压定理会带来高于Gardner基态的能量约束，提出格罗莫夫基态的核心难题并推测其线性形式可解。
We prove that the non-squeezing theorem of Gromov holds for symplectomorphisms on an infinite-dimensional symplectic Hilbert space, under the assumption that the image of the ball is convex. The proof is based on the construction by duality methods of a symplectic capacity for bounded convex neighbourhoods of the origin. We also discuss some examples of symplectomorphisms on infinite-dimensional spaces exhibiting behaviours which would be impossible in finite dimensions.
We prove a non-squeezing result for infinite-dimensional Hamiltonian flows using non-standard model theory. For this we prove the existence of a corresponding family of pseudoholomorphic spheres and characterize the maximal time in terms of a limiting behaviour for these curves. While our proof is based on the finite-dimensional results from Gromov's original proof, we do not ask for any prior knowledge of non-standard model theory.
We develop the theory of $J$-holomorphic discs in Hilbert spaces with almost complex structures. As an aplication, we prove a version of Gromov's symplectic non-squeezing theorem for Hilbert spaces. It can be applied to short-time symplectic flows of a wide class of Hamiltonian PDEs.
Rigidity and flexibility in $p$-adic symplectic geometry
本文研究 $p$-进辛几何中的非挤压问题，证明一般情形具有柔性，但仿射及环面等变非线性嵌入仍具刚性，并引入等变 $p$-进解析辛容量。
Gromov's famous non-squeezing theorem (1985) states that the standard symplectic ball cannot be symplectically squeezed into any cylinder of smaller radius. Does there exist an analogue of this result in contact geometry? Our main finding is that the answer depends on the sizes of the domains in question: We establish contact non-squeezing on large scales, and show that it disappears on small scales. The algebraic counterpart of the (non)-squeezing problem for contact domains is the question of existence of a natural partial order on the universal cover of the contactomorphisms group of a cont…
Non-squeezing and capacities for some calibrated geometries
本文改进并证明了关于辛形式幂的仿射非挤压定理，并研究了校准形式（特别是特殊拉格朗日情形）的仿射非挤压定理，建立了仿射特殊拉格朗日版本的刚性定理。
Pinwheels in symplectic rational and ruled surfaces and non-squeezing of rational homology balls
本研究利用殆环纤维化与辛有理爆发方法，确定阈限型拉格朗日风车的嵌入条件，否定回答Kronheimer相关问题，推广Gromov非挤压定理，并证明拉格朗日风车的多项独立性质。
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