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### 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?

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 (...

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### 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 ...

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### 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 $l$?

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### 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 ...

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### 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 ...

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### 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 ...