Questions tagged [systolic-geometry]
The systolic-geometry tag has no usage guidance.
19
questions
2
votes
0
answers
149
views
Upper bound of special Cheeger constant on $(S^2,g)$
$(S^2,g)$ is 2-dimensional sphere with Riemannian metric.The Cheeger constant of $(S^2,g)$ is
$$
h(S^2,g)=\inf_{\gamma} \frac{|\gamma|_g}{\min\{|A_1|_g, |A_2|_g\}}
$$
take the infimum over all closed ...
5
votes
0
answers
340
views
Extending Gromov's inequality
In 1981 Gromov proved that all Riemannian metrics on the complex projective space $\mathbb CP^n$ satisfy the bound
$$\DeclareMathOperator{stsys}{stsys} \DeclareMathOperator{vol}{vol}
\frac{\stsys_2^n}{...
16
votes
1
answer
977
views
Status of Larry Guth's Sponge Problem
[Edited Jan 23, 2021]
Let $D^n$ be the $n$-dimensional unit radius disk in euclidean $\mathbb{R}^n$.
Larry Guth's Sponge Problem asks: Does there exist a constant $\epsilon=\epsilon_n$ such that every ...
9
votes
0
answers
235
views
Systole of Riemann surfaces of genus $g$
In Buser and Sarnak's "On the period matrix of a Riemann surface
of large genus", we get
$$\frac4{3}\le\limsup_{g\rightarrow\infty}\frac{\max\{\operatorname{sys}(S)|S\in\mathcal{M}_g\}}{\log ...
10
votes
2
answers
732
views
On Gromov's proof of the systolic inequality $\operatorname{Sys}_1(M)\leq 6\operatorname{FillRad}(M)$
In the page 10 of the paper "Filling Riemannian manifolds" by Gromov (ProjetEuclid link), the author proves the following inequality (1.2) relating the systole and the filling radius of manifolds.
$$\...
6
votes
1
answer
163
views
Is the volume of relative cycles at least the systole of the manifold?
Let $M$ be a manifold with boundary $\partial M$. Suppose that $M$ is equipped with some structure for which a notion of volume for chains can be defined. For example, if $M$ is triangulated, then the ...
6
votes
2
answers
648
views
Flat metrics on $n$-toruses, their systoles, and the "shortest vector problem"
Apologies if this is too basic, but I haven't been able to find any info about the question. Is there anything known about the moduli space of flat metrics on an $n$-torus (i.e., $(S^1)^n$)? ...
12
votes
1
answer
773
views
Finite covers of hyperbolic surfaces and the `second systole´
We are interested in the following ´relative´ version of residual finiteness for fundamental groups of surfaces. Similar discussions where given in this question: injectivity radius of hyperbolic ...
1
vote
0
answers
91
views
Lower bound for $k$-systole on $T^{2k}$ using Euclidean metric
Consider the torus ${\mathbb T}^{2k}={\mathbb R}^{2k}/\Lambda$, where $\Lambda$ is a lattice. I am looking for a lower bound on the volume using the Euclidean metric of a cycle that represents a ...
1
vote
1
answer
134
views
Upper bounds for systoles on punctured surfaces
If the systole is defined as the length of the shortest essential simple closed curve are there any known upper bounds for hyperbolic surfaces with punctures?
8
votes
0
answers
280
views
Systoles of hyperbolic (Riemann) surfaces of large genus
Let $m$ be a Riemannian metric on $S_g$ the surface of genus $g$, and $sys(m)$ be the length of the shortest non contractible cycle with respect to $m$.
The systolic inequality claims that for any ...
5
votes
1
answer
217
views
Stability of Pu's isosystolic inequality
The volume of a Riemannian metric on the projective plane is $2\pi$ and length of every non-contractible loop is greater than $\pi - \epsilon$ for some small, positive number $\epsilon$. Is this ...
24
votes
3
answers
1k
views
Isoperimetric inequality on a Riemannian sphere
Consider a two-dimensional sphere with a Riemannian metric of total area $4\pi$. Does there exist a subset whose area equals $2\pi$ and whose boundary has length no greater than $2\pi$?
(To avoid ...
2
votes
1
answer
458
views
Is there a lower bound for variance in terms of curvature?
If the Gaussian curvature of the metric $g= f^2(x,y)(dx^2+dy^2)$ is nonzero then $f$ cannot be constant. This can be expressed by stating that the (probabilistic) variance $Var(f)$ of $f$ is nonzero (...
5
votes
1
answer
310
views
Fixing a proof of the systolic inequality for higher genus surfaces
I'm currently learning some stuffs about systolic inequalities. While reading the relevant sections (p329 to 340) in Berger's Panoramic View of Riemannian Geometry, I noticed a gap in one of the ...
6
votes
2
answers
1k
views
injectivity radius of hyperbolic surface
Given a positive real number $l$. Does there exist a closed hyperbolic surface $X$ so that injectivity radius not less than $l$?
1
vote
1
answer
194
views
systole and residualy finite fundamental group
it is well known that if the fundamental group of a manifold M is residualy finite then for every point p in M and every epsilon positive there is a finite covering such that for a point q in the ...
2
votes
0
answers
241
views
Reverse engineering a metric from its properties
I have been reading an article, and I am not clear on what he does in this one part.
He starts with a Riemmannian manifold $\Sigma$ that is $\mathbb{H}^2/\Gamma$, a quotient of the hyperbolic space ...
4
votes
1
answer
610
views
What would a graduate course on systolic geometry typically cover?
It's all in the title basically. There's an interesting topic called systolic geometry that has grown a lot in the past 30 years, with a (first?) textbook on the subject by M.Katz (AMS 2007).
So I ...