Bounding volume by systoles of 3-manifolds

$\mathbb{Z}_{2}$-coefficient homology $(1, 2)$-systolic freedom of $\mathbb{R}\mathbb{P}^{3}$ # $\mathbb{R}\mathbb{P}^{3}$

We prove the $3$-manifold $\RP^3 \# \RP^3$ is of $\Z_{2}$-coefficient homology $(1, 2)$-systolic freedom. Given a Riemannian metric on $\RP^{3}\# \RP^{3}$, we define $\Z_{2}$-coefficient homology $1$-systole as the infimum of lengths of all nonseparating geodesic loops representing nontrivial classes in $H_{1}(\RP^3\#\RP^3; \Z_{2})$. The $\Z_{2}$-coefficient homology $2$-systole is defined to be the infimum of areas of all nonseparating surfaces representing nontrivial classes in $H_{2}(\RP^{3}\#\RP^{3}; \Z_2)$. In the paper we show that there exists a sequence of Riemannian metrics on $\RP^{3…

Systolic freedom of orientable manifolds

In 1972, Marcel Berger defined a metric invariant that captures the `size' of k-dimensional homology of a Riemannian manifold. This invariant came to be called the k-dimensional SYSTOLE. He asked if the systoles can be constrained by the volume, in the spirit of the 1949 theorem of C. Loewner. We construct metrics, inspired by M. Gromov's 1993 example, which give a negative answer for large classes of manifolds, for the product of systoles in a pair of complementary dimensions. An obstruction (restriction on k modulo 4) to constructing further examples by our methods seems to reside in the fre…

$\u2102P^{n}$中全纯链的等收缩不等式

Isosystolic Inequalities for Holomorphic Chains in $\mathbb{C}P^{n}$

该研究定义了复射影空间上埃尔米特度量的全纯k-收缩，证明Fubini-Study度量是Gauduchon度量中体积归一化全纯$(n-1)$-收缩的局部极小值，并构造出接近Fubini-Study度量、同调2-收缩由非全纯链实现的Gauduchon度量。

On manifolds satisfying stable systolic inequalities

We show that for closed orientable manifolds the $k$-dimensional stable systole admits a metric-independent volume bound if and only if there are cohomology classes of degree $k$ that generate cohomology in top-degree. Moreover, it turns out that in the nonorientable case such a bound does not exist for stable systoles of dimension at least two. Additionally, we prove that the stable systolic constant depends only on the image of the fundamental class in a suitable Eilenberg-Mac Lane space. Consequently, the stable $k$-systolic constant is completely determined by the multilinear intersection …

Sur la géométrie systolique des variétés de Bieberbach

The systole of a compact non simply connected Riemannian manifold is the smallest length of a non-contractible closed curve ; the systolic ratio is the quotient $(\mathrm{systole})^n/\mathrm{volume}$. Its supremum on the set of all the riemannian metrics, is known to be finite for a large class of manifolds, including the $K(π,1)$. We study the optimal systolic ratio of compact, 3-dimensional non orientable Bieberbach manifolds, and prove that it cannot be realized by a flat metric.

Kissing numbers of closed hyperbolic manifolds

We prove an upper bound for the number of shortest closed geodesics in a closed hyperbolic manifold of any dimension in terms of its volume and systole, generalizing a theorem of Parlier for surfaces. We also obtain bounds on the number of primitive closed geodesics with length in a given interval that are uniform for all closed hyperbolic manifolds with bounded geometry. The proofs rely on the Selberg trace formula.

Margulis Lemma, entropy and free products

We prove a Margulis' Lemma à la Besson Courtois Gallot, for manifolds whose fundamental group is a nontrivial free product A*B, without 2-torsion. Moreover, if A*B is torsion-free we give a lower bound for the homotopy systole in terms of upper bounds on the diameter and the volume entropy. We also provide examples and counterexamples showing the optimality of our assumption. Finally we give two applications of this result: a finiteness theorem and a volume estimate for reducible manifolds.

Local topological rigidity of non-geometric $3$-manifolds

We study Riemannian metrics on compact, torsionless, non-geometric $3$-manifolds, i.e. whose interior does not support any of the eight model geometries. We prove a lower bound "à la Margulis" for the systole and a volume estimate for these manifolds, only in terms of an upper bound of entropy and diameter. We then deduce corresponding local topological rigidy results in the class $\mathscr{M}_{ngt}^\partial (E,D) $ of compact non-geometric 3-manifolds with torsionless fundamental group (with possibly empty, non-spherical boundary) whose entropy and diameter are bounded respectively by $E, D$.…

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…

利用标量曲率下界和等周常数界定 $\widehat{A}$ 亏格

Bounding the A-hat genus using scalar curvature lower bounds and isoperimetric constants

本文利用标量曲率下界、Neumann 等周常数和体积，通过 Dirac 算子谱分析证明了光滑闭 spin Riemannian 流形 $\widehat{A}$ 亏格的上界，并构造例子说明该常数必要性，部分回答了 Gromov 的问题。

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…