Questions tagged [systolic-geometry]

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15
votes
1answer
696 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 ...
7
votes
0answers
102 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 ...
8
votes
2answers
542 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. $$\...
5
votes
1answer
138 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
2answers
357 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
1answer
604 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
0answers
70 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
1answer
122 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
0answers
264 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
1answer
199 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
3answers
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
1answer
371 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
1answer
294 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
2answers
822 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
1answer
189 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
0answers
232 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
1answer
593 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 ...