## Of Loewner and Besicovitch

Posted in Math by pgadey on 2013/11/05

I’d like to share some of the notes that I’m writing up about systoles. After a little bit of preliminaries we’ll see a slick proof the systolic inequality in the torus case.

The systole of manifold is the length of the shortest non-contractible curve in the manifold. Systoles hard to estimate. In general there are many many non-contractible curves, and its not easy to track down which one should be smallest. If someone hands you a donut, you’ll visually guess the systole correctly. If someone hands you a coffee cup, it’s still clear. Once you get a generic metric, you’re in deep water. Loewner‘s theorem gives us an upper bound on the systole a Riemannian 2-torus (generalized donut / coffee cup case).

Definition 1 Given a connected closed Riemannian ${n}$-manifold ${(M,g)}$ the systole of ${M}$ is ${\mathop{\operatorfont sys}\nolimits(M) = \inf \{\mathop{\operatorfont vol}\nolimits_1(\gamma) : [\gamma] \neq 0 \in \pi_1(M) \}}$. That is, the systole is the length of the shortest non-contractible curve in ${M}$.

Theorem 2 If ${(T^2, g)}$ is a Riemannian 2-torus then:

$\displaystyle \mathop{\operatorfont sys}\nolimits_1(T^2) \leq \sqrt{ \frac{2}{\sqrt{3} }\mathop{\operatorfont vol}\nolimits_2(T) }$

Moreover, equality occurs if and only if ${(T^2,g)}$ is the flat torus ${{\mathbb C} / (1{\mathbb Z} \oplus \omega {\mathbb Z})}$ where ${\omega}$ is a third root of unity.

The equality case described above is worth exploring. Let ${T}$ denote the torus described in the equality case. We take the plane, here written as ${{\mathbb C}}$ and we mod out by a particular lattice: ${1{\mathbb Z} \oplus \omega{\mathbb Z}}$. This lattice has a fundamental domain the parallelogram bounded by ${0, 1, \omega}$, and ${1+\omega}$. We then get that ${\mathop{\operatorfont vol}\nolimits_2(T) = 1 \cdot \frac{ \sqrt{3} }{ 2 }}$. Thus, the upper bound in Loewner’s theorem in one. We certainly have many curves which realize this bound. Any curve parallel to the edge ${\{0,1\}}$ or the edge ${\{0, \omega\}}$ will do. This shows ${\mathop{\operatorfont sys}\nolimits_1(T) \leq 1}$ we now check that one is the shortest length of a non-contractible curve. We’ll look at things in ${{\mathbb C}}$, the universal cover of ${T}$. Consider a curve ${\gamma}$ starting from the origin of length less than one. If ${\gamma}$ is non-contractible then it has to be closed and it can’t end at the origin, since otherwise we could contract it. The nearest points in ${{\mathbb C}}$ that get identified with the origin are ${\pm 1, \pm \omega}$. One can see that to hit such a point, ${\gamma}$ must have length at least one. Thus, there are no non-contractible curves of length less than one. Thus equality holds in Loewner’s inequality. Checking that the lattice described above maximizes the systole volume ratio is a good exercise in flat geometry.

We’re going to outline a proof that doesn’t give us the sharp constant, but which is quite instructive and natural. The reference for this is Gromov’s wonderful book: Metric Structures. First we introduce the following useful quantitative lemma:

Lemma 3 (Besicovitch) Let ${I = [0,1]}$ denote a closed unit interval and let ${S = (I^2, g)}$ be a Riemannian square. Let ${d_1 = d(\{0\} \times I, \{1\} \times I)}$ and ${d_2 = d(I \times \{0\}, I \times \{1\})}$ denote the distances between pairs of opposite sides. One has:

$\displaystyle \mathop{\operatorfont vol}\nolimits_2(S) \geq d_1 d_2$

Now we’ll deduce Theorem~2 (without the sharp constant) from the lemma above.

Proof:

First note that if we cut ${T}$ along the shortest non-seperating curve we obtain two boundary components ${\gamma_1, \gamma_2}$. Let ${\gamma}$ be the shortest curve homotopic to one of these boundary components. In particular, ${\mathop{\operatorfont vol}\nolimits_1(\gamma) = \mathop{\operatorfont sys}\nolimits_1(\gamma)}$. We apply the co-area formula. We have ${\mathop{\operatorfont vol}\nolimits_2(T) = \int_0^r d^{-1}(t) dt \geq \mathop{\operatorfont vol}\nolimits_1(\gamma) d(\gamma_1,\gamma_2)}$. So, ${\mathop{\operatorfont vol}\nolimits_1(\gamma) \leq \mathop{\operatorfont vol}\nolimits_2(T) / d(\gamma_1, \gamma_2)}$. This is quite nice. Pick an arc ${\tilde{\alpha}}$ from ${\gamma_1}$ to ${\gamma_2}$ realizing ${d(\gamma_1,\gamma_2)}$. Note that its end points might not be equal and so we complete it to a curve ${\alpha}$ by adding in an arc of ${\gamma}$ of length at most ${\frac{ 1 }{ 2 } \mathop{\operatorfont vol}\nolimits_1(\gamma)}$. We then have ${\mathop{\operatorfont vol}\nolimits_1(\alpha) \leq d(\gamma_1, \gamma_2) + \frac{ 1 }{ 2 } \mathop{\operatorfont vol}\nolimits_1(\gamma)}$.

$\displaystyle d(\gamma_1, \gamma_2) + \frac{ 1 }{ 2 } \mathop{\operatorfont vol}\nolimits_1(\gamma) \geq \mathop{\operatorfont vol}\nolimits_1(\alpha) \geq \mathop{\operatorfont vol}\nolimits_1(\gamma) \Rightarrow 2 d(\gamma_1, \gamma_2) \geq \mathop{\operatorfont vol}\nolimits_1(\gamma)$

We then apply Besicovitch and obtain:

$\displaystyle \mathop{\operatorfont vol}\nolimits_2(T) \geq \mathop{\operatorfont vol}\nolimits_1(\gamma) d(\gamma_1,\gamma_2) \geq \frac{ 1 }{ 2 } \left( \mathop{\operatorfont vol}\nolimits_1(\gamma_1) \right)^2$

$\Box$

Eventually I’ll get to writing about what this is useful for and how it’s been generalized. Loewner’s systolic inequality inspired a lot of good work. For much more inspired and in-depth introduction to systolic inequalities one can do no better than Larry Guth‘s article: Metaphors in systolic Geometry.

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