Metaphors in systolic geometry

Cohomological dimension, self-linking, and systolic geometry

Given a closed manifold M, we prove the upper bound of (n+d)/2 for the length of a product of systoles that can form a curvature-free lower bound for the total volume of M, in the spirit of M. Gromov's systolic inequalities. Here n is the dimension of M, while d is the is the cohomological dimension of its fundamental group. We apply this upper bound to show that, in the case of a 4-manifold, the Lusternik--Schnirelmann category is an upper bound for such length. Furthermore we prove a systolic inequality on a manifold M with b_1(M)=2 in the presence of a nontrivial self-linking class of the t…

质量和收缩期几何结构的局部校准

Local calibration of mass and systolic geometry

本文证明了单连通n-流形X的同时（k，n-k）-收缩自由度，其中相邻整数对k小于n/2.我们的建设，涉及到最近的结果，我。Babenko集中在X的合适的k维子流形的邻域中。我们采用在这样的社区支持的微分形式的校准，提供下界的（n-k）-systoles。同时，尽管整体内射性半径为零，k-系统仍由单调性公式和构造在适当的（n-k+1）维子流形邻域中的有界几何从下而上控制.该结构是几何的，代数拓扑成分减少到庞加莱对偶和托姆定理表示多个同调类的子流形。本文的结果与A.利用有理同伦理论，讨论了X的k收缩自由度。我们的研究结果收缩自由度对比存在稳定的收缩不等式，研究在联合工作与V。

三维中S¹不变切触形式的收缩不等式

Systolic inequalities for S1-invariant contact forms in dimension three

该研究在切触几何领域，针对欧拉数非零的Seifert丛，证明了适用于所有底层圆周作用不变切触形式的通用收缩不等式，即用切触体积一致上界闭Reeb轨道的最短周期。

Infinitesimal Systolic Rigidity of Metrics all of whose Geodesics are Closed and of the same Length

The results of this paper have been greatly superseded by those in the paper "Contact geometry and isosystolic inequalities" (arXiv:1109.4253) by the same authors.

Contact geometry and isosystolic inequalities

A long-standing open problem in systolic geometry asks whether a Riemannian metric on the real projective space whose volume equals that of the canonical metric, but is not isometric to it, must necessarily carry a periodic geodesic of length smaller than π. A contact-geometric reformulation of systolic geometry and the use of canonical perturbation theory allow us to solve a parametric version of this problem. Namely, we show that if g_s is a smooth volume-preserving deformation of the canonical metric and at s=0 the deformation is not tangent to all orders to trivial deformations (i.e., to d…

A local systolic-diastolic inequality in contact and symplectic geometry

Let $Σ$ be a connected closed three-manifold, and let $t_Σ$ be the order of the torsion subgroup of $H_1(Σ;\mathbb Z)$. For a contact form $α$ on $Σ$, we denote by $\mathrm{Volume}(α)$ the contact volume of $α$, and by $T_{\min}(α)$ and $T_{\max}(α)$ the minimal period and the maximal period of prime periodic orbits of the Reeb flow of $α$ respectively. We say that $α$ is Zoll if its Reeb flow generates a free $S^1$-action on $Σ$. We prove that every Zoll contact form $α_*$ on $Σ$ admits a $C^3$-neighbourhood $\mathcal U$ in the space of contact forms such that \[ t_ΣT_{\min}(α)^2\leq \mathrm{…

小分离器、l^∞-宽度的上界与 systole 几何

Small separators, upper bounds for $l^\infty$-widths, and systolic geometry

该研究探讨闭子流形的欧氏体积与$l^\infty$-宽度对维度的依赖，推导两类$l^\infty$-宽度上界，作为推论得到本质子流形的非可缩闭曲线存在性，还引入了 systole 几何的非线性新方法。