Free abelian covers, short loops, stable length, and systolic inequalities

Systolic inequalities and minimal hypersurfaces

We give a short proof of the systolic inequality for the n-dimensional torus. The proof uses minimal hypersurfaces. It is based on the Schoen-Yau proof that an n-dimensional torus admits no metric of positive scalar curvature.

关于comass和稳定扭率不等式

On comass and stable systolic inequalities

本文改进了comass范数与欧几里得范数的比值上界，并证明了在特定条件下流形的稳定扭率不等式。

Lusternik-Schnirelmann category and systolic category of low dimensional manifolds

We show that the geometry of a Riemannian manifold (M,g) is sensitive to the apparently purely homotopy-theoretic invariant of M known as the Lusternik-Schnirelmann category, denoted cat_{LS}(M). Here we introduce a Riemannian analogue of cat_{LS}(M), called the systolic category of M. It is denoted cat_{sys}(M), and defined in terms of the existence of systolic inequalities satisfied by every metric g, as initiated by C. Loewner and later developed by M. Gromov. We compare the two categories. In all our examples, the inequality cat_{sys}(M) \le cat_{LS}(M) is satisfied, which typically turns …

Sharp systolic inequalities for rotationally symmetric 2-orbifolds

We show that suitably defined systolic ratios are globally bounded from above on the space of rotationally symmetric spindle orbifolds and that the upper bound is attained precisely at so-called Besse metrics, i.e. Riemannian orbifold metrics all of whose geodesics are closed.

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。

Combinatorial systolic inequalities

We establish combinatorial versions of various classical systolic inequalities. For a smooth triangulation of a closed smooth manifold, the minimal number of edges in a homotopically non-trivial loop contained in the $1$-skeleton gives an integer called the combinatorial systole. The number of top-dimensional simplices in the triangulation gives another integer called the combinatorial volume. We show that a class of smooth manifolds satisfies a systolic inequality for all Riemannian metrics if and only if it satisfies a corresponding combinatorial systolic inequality for all smooth triangulat…

E_7, Wirtinger inequalities, Cayley 4-form, and homotopy

We study optimal curvature-free inequalities of the type discovered by C. Loewner and M. Gromov, using a generalisation of the Wirtinger inequality for the comass. Using a model for the classifying space BS^3 built inductively out of BS^1, we prove that the symmetric metrics of certain two-point homogeneous manifolds turn out not to be the systolically optimal metrics on those manifolds. We point out the unexpected role played by the exceptional Lie algebra E_7 in systolic geometry, via the calculation of Wirtinger constants. Using a technique of pullback with controlled systolic ratio, we cal…

First steps into the world of systolic inequalities: From Riemannian to symplectic geometry

Our aim is to give a friendly introduction for students to systolic inequalities. We will stress the relationships between the classical formulation for Riemannian metrics and more recent developments related to symplectic measurements and the Viterbo conjecture. This will give us a perfect excuse to introduce the reader to some important ideas in Riemannian and symplectic geometry.

Systolic volume and minimal entropy of aspherical manifolds

We establish isosystolic inequalities for a class of manifolds which includes the aspherical manifolds. In particular, we relate the systolic volume of aspherical manifolds first to their minimal entropy, then to the algebraic entropy of their fundamental groups.

Higher systolic inequalities for 3-dimensional contact manifolds

A contact form is called Besse when the associated Reeb flow is periodic. We prove that Besse contact forms on closed connected 3-manifolds are the local maximizers of suitable higher systolic ratios. Our result extends earlier ones for Zoll contact forms, that is, contact forms whose Reeb flow defines a free circle action.

Extremal length in higher dimensions and complex systolic inequalities

Extremal length is a classical tool in 1-dimensional complex analysis for building conformal invariants. We propose a higher-dimensional generalization for complex manifolds and provide some ideas on how to estimate and calculate it. We also show how to formulate certain natural geometric inequalities concerning moduli spaces in terms of a complex analogue of the classical Riemannian notion of systole.

Sharp systolic inequalities for Riemannian and Finsler spheres of revolution

We prove that the systolic ratio of a sphere of revolution $S$ does not exceed $π$ and equals $π$ if and only if $S$ is Zoll. More generally, we consider the rotationally symmetric Finsler metrics on a sphere of revolution which are defined by shifting the tangent unit circles by a Killing vector field. We prove that in this class of metrics the systolic ratio does not exceed $π$ and equals $π$ if and only if the metric is Riemannian and Zoll.

Stable systolic inequalities and cohomology products

Multiplicative relations in the cohomology ring of a manifold impose constraints upon its stable systoles. Given a compact Riemannian manifold (X,g), its real homology H_*(X,R) is naturally endowed with the stable norm. Briefly, if h\in H_k(X,R) then the stable norm of h is the infimum of the Riemannian k-volumes of real cycles representing h. The stable k-systole is the minimum of the stable norm over nonzero elements in the lattice of integral classes in H_k(X,R). Relying on results from the geometry of numbers due to W. Banaszczyk, and extending work by M. Gromov and J. Hebda, we prove metr…

莫比乌斯带与克莱因瓶收缩不等式稳定性

Stability of systolic inequalities for the Möbius strip and Klein bottle

本文研究莫比乌斯带和克莱因瓶上收缩面积不等式的稳定性，给出Bavard结果的替代证明，并用共形因子到极小度量的L²距离估计收缩缺陷。

Positive loops and $L^{\infty}$-contact systolic inequalities

We prove an inequality between the $L^{\infty}$-norm of the contact Hamiltonian of a positive loop of contactomorphims and the minimal Reeb period. This implies that there are no small positive loops on hypertight or Liouville fillable contact manifolds. Non-existence of small positive loops for overtwisted 3-manifolds was proved by Casals-Presas-Sandon in [CPS16]. As corollaries of the inequality we deduce various results. E.g. we prove that certain periodic Reeb flows are the unique minimizers of the $L^\infty$-norm. Moreover, we establish $L^\infty$-type contact systolic inequalities in the…

Volume of Riemannian manifolds, geometric inequalities, and homotopy theory

We outline the current state of knowledge regarding geometric inequalities of systolic type, and prove new results, including systolic freedom in dimension 4. Namely, every compact, orientable, smooth 4-manifold X admits metrics of arbitrarily small volume such that every orientable, immersed surface of smaller than unit area is necessarily null-homologous in X.

通过模n覆盖实现稳定的周长不等式

Stable systolic inequalities via mod n covering

本文提出了一种基于模n覆盖的方法来研究稳定的周长不等式，通过强制特定的上积或指标非零，找到短的整数提升，特别是在秩为二的情况下可以计算覆盖常数。该方法在无曲率应用中改进了S²×S²的稳定二周长界为2，并适用于所有具有b₂=2的定向四维流形。在正标量曲率下界条件下，结合尖锐的线丛不等式，得到稳定的二周长界，从而为奇数复射影空间给出精确的稳定二周长不等式，并为(S²)^m在标量曲率至少为2m时给出O(m log m)的界。

推广 Gromov 的最优收缩不等式

Extending Gromov's optimal systolic inequality

本文将 Gromov 针对复射影空间的最优收缩不等式推广到基本上同调类由二维类的 cup 积构成的流形。

通过填充极小性得到度量曲面的收缩不等式

Systolic inequalities for metric surfaces via filling minimality

研究与亏格为1的环面或实射影平面同胚的长度空间的最优收缩不等式，通过分析万有覆盖渐近体积增长等方法，推广了黎曼和芬斯勒曲面的经典结果，证明依赖相关理论并基于前人工作。

通过稳定性条件研究K3曲面的收缩不等式

Systolic inequalities for K3 surfaces via stability conditions

本文引入Bridgeland稳定性条件的范畴收缩与范畴体积概念，证明任意射影K3曲面上存在仅依赖Picard群秩与判别式的常数C，使收缩平方不超过C倍体积，推广了二维环面经典收缩不等式并讨论其在辛几何中的应用。

S²上主S¹丛上不变紧接触形式的尖锐收缩不等式

Sharp systolic inequalities for invariant tight contact forms on principal S1-bundles over S2

本文研究S²上非平凡主S¹丛中S¹不变紧接触形式的收缩问题，证明了依赖Euler类的尖锐不等式，并将其应用于旋转对称Finsler度量、可缩Reeb轨道及Viterbo猜想特例。

收缩不等式与Horowitz-Myers猜想

Systolic inequalities and the Horowitz-Myers conjecture

对于3至7维紧流形，在标量曲率下界条件下，建立了边界平均曲率下确界与边界收缩之间的关系，并由此得到新的正能量定理，等号由Horowitz-Myers度规实现。

Sharp systolic inequalities for Reeb flows on the three-sphere

The systolic ratio of a contact form $α$ on the three-sphere is the quantity \[ ρ_{\mathrm{sys}}(α) = \frac{T_{\min}(α)^2}{\mathrm{vol}(S^3,α\wedge dα)}, \] where $T_{\min}(α)$ is the minimal period of closed Reeb orbits on $(S^3,α)$. A Zoll contact form is a contact form such that all the orbits of the corresponding Reeb flow are closed and have the same period. Our first main result is that $ρ_{\mathrm{sys}}\leq 1$ in a neighbourhood of the space of Zoll contact forms on $S^3$, with equality holding precisely at Zoll contact forms. This implies a particular case of a conjecture of Viterbo, a…

Kähler流形的紧系数不等式

Sharp systolic inequalities for Kähler manifolds

本文研究了具有正标量曲率度量的二维系数不变量的紧系数不等式，包括紧Kähler流形的2-系数和球面2-系数，以及一般spin^c流形及其积上的稳定2-系数。这些界限在复射影空间配备Fubini-Study度量时达到等号，并进一步细化Fano流形以区分复二次曲面、三次曲面和四次曲面。此外，还获得了允许Kähler度量总标量曲率非负的流形的代数特征化，从而证明了Gromov的Kähler度量有理本质性猜想。最后，证明了在一般本质性条件下spin^c流形的稳定2-系数和Kähler流形的Gromov宽度、体积和更高稳定系数的统一界限。

源自辛嵌入的球面上的收缩不等式

Systolic inequalities on the sphere from symplectic embeddings

本文利用 Hutchings 的辛容量估计 $T^*S^2$ 中纤维星形超曲面的 Reeb 轨道最小作用量，并引入纤维 $β$-平衡概念，证明相关收缩上界及对跖对称下 $δ$-挤扁度量的非精确估计。

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…

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.

A systolic inequality for geodesic flows on the two-sphere

For a Riemannian metric $g$ on the two-sphere, let $\ell_{\min}(g)$ be the length of the shortest closed geodesic and $\ell_{\max}(g)$ be the length of the longest simple closed geodesic. We prove that if the curvature of $g$ is positive and sufficiently pinched, then the sharp systolic inequalities \[ \ell_{\rm min}(g)^2 \leq π\ {\rm Area}(S^2,g) \leq \ell_{\max}(g)^2, \] hold, and each of these two inequalities is an equality if and only if the metric $g$ is Zoll. The first inequality answers positively a conjecture of Babenko and Balacheff. The proof combines arguments from Riemannian and s…

Kähler爆破上的纯量曲率与收缩不等式

Scalar curvature on Kähler blow-ups and systolic inequalities

本文发展针对特定Kähler流形的加权水平集方法，证明其爆破的Kähler度量序列的纯量曲率逼近原度量，建立正纯量曲率Kähler流形的精确2-收缩估计及偶收缩不等式。

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…

Equivariant Holomorphic Morse Inequalities II: Torus and Non-Abelian Group Actions

We extend the equivariant holomorphic Morse inequalities of circle actions to cases with torus and non-Abelian group actions on holomorphic vector bundles over Kahler manifolds and show the necessity of the Kahler condition. For torus actions, there is a set of inequalities for each choice of action chambers specifying directions in the Lie algebra of the torus. We apply the results to invariant line bundles over toric manifolds. If the group is non-Abelian, there is in addition an action of the Weyl group on the fixed-point set of its maximal torus. The sum over the fixed points can be rearra…

Arakelov不等式与$\mathcal{A}_g$中全测地球商的特征刻画

Arakelov inequalities and characterization of totally geodesic ball quotients in $\mathcal{A}_g$

本文建立主极化复阿贝尔簇族的Arakelov不等式，在等式情形给出阿贝尔簇模空间中全测地球商的数值刻画，去除了Möller-Viehweg-Zuo的强正性条件，并在紧致基与拟射影曲面情形分别给出新证明。