Parker Glynn-Adey

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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