Tag: length

Longest shortest geodesic on a 2-sphere

ConanLeave a comment

This is a little note about constructing a Riemannian 2-sphere which has longer shortest geodesic than the round 2-sphere of same area.

—–  Background Story  —–

So there has been this thing called ‘mathematical conversations’ at the IAS, which involves someone present a topic that’s elementary enough to be accessible to mathematicians in all fields and yet can be expanded in different directions and lead into interesting interdisciplinary discussions.

Nancy Hingston gave one of those conversations about simple geodesics on the two-sphere one night and I was (thanks to Maria Trnkova who dragged me in) able to attend.

So she talked about some fascinating history of proving the existence of closed geodesics and later simple closed geodesics on generic Riemannian two-spheres.

Something about this talk obviously touched my ‘systolic nerve’, so when the discussion session came up I asked whether we have bounds on ‘length of longest possible shortest closed geodesic on a sphere with unit area’. The question seem to have generated some interest in the audience and resulted in a back-and-forth discussion (some of which I had no clue what they were talking about). So the conclusion was at least nobody knows such a result on top of their head and perhaps optimum is obtained by the round sphere.

—–  End of Story —-

A couple of post-docs caught me afterwards (Unfortunately I didn’t get their names down, if you happen to know who they are, tell me~) and suggested that suspending a smooth triangular region and smoothen the corners might have longer shortest geodesic than the round sphere:

The evidence being the fact that on the plane a rounded corner triangular contour has larger ‘width’ than the disc of same area. (note such thing can be made to have same width in all directions)

Well that’s pretty nice, so I went home and did a little high-school computations. The difficulty about the pillow is that the shortest geodesic isn’t necessarily the one that goes through the ‘tip’ and ‘mid-point of the base’, something else might be shorter. I have no idea how to argue that.

A suspicious short geodesic:

So I ended up going with something much simpler, namely gluing together two identical copies of the flat equilateral triangles. This can be made to a Riemannian metric by smoothing the edge and corners a little bit:

Okay, now the situation is super simple~ I want to prove that this ‘sphere’ (let’s call it S from now on) has shortest geodesic longer than the round sphere (\mathbb{S}^2)!

Of course we suppose both S and \mathbb{S}^2 has area 1.

Claim: The shortest geodesic on S has length \sqrt[4]{12} (which is length of the one through the tip and mid-point of the opposite edge.)

Proof: The shortest closed geodesic passing through the corner is the one described above, since any other such geodesics must contain two symmetric segments from the corner to the bottom edge on the two triangles, those two segments alone is longer than the one orthogonal to the edge.

That middle one has length 2h where

A(\Delta) = 1/2 = h^2/\sqrt{3}

i.e. h = \sqrt[4]{3} / \sqrt{2}, \ \ell = 2h = \sqrt[4]{12}

The good thing about working with flat triangles is that now I know what the closed geodesics are~

First we observe any closed geodesic not passing through the corner is a periodical billiard path in the triangular table with even period.

So let’s ‘unfold’ the triangles on the plane. Such periodic orbits correspond to connecting two corresponding points on a pair of identified parallel edges and the segment between them intersecting an even number of tiles.

W.L.O.G we assume the first point in on edge e. Since we are interested in orbits having shortest length, let’s take neighborhood of radius \sqrt[4]{12} + \epsilon around our edge e: (all edges with arrows are identified copies of e)

There are only 6 parallel copies of e in the neighborhood:

Note that no matter what point p on e we start with, the distance from p to another copy of it on any of the six edges is EQUAL to \sqrt[4]{12}. (easy to see since one can slide the segments to begin and end on vertices.)

Hence we conclude there are no shorter periodic billiard paths, i.e. the shortest closed geodesic on S has length \sqrt[4]{12}.

Note it’s curious that there are a huge amount of closed geodesics of that particular length, most of them are not even simple! However it seems that after we smoothen S to a Riemannian metric, the non-simple ones all become a little longer than that simple one through the corner. I wonder if it’s possible that on a Riemannian sphere the shortest closed geodesic is a non-simple one.

Anyways, now let’s return to \mathbb{S}^2~ So the surface area is 1 hence the radius is r= \sqrt{1/4\pi} = \frac{1}{2\sqrt{\pi}}

Any closed geodesics is a multiple of a great circle, hence the shortest geodesic has length 2 \pi r = \sqrt{\pi}, which is just slightly shorter than \sqrt[4]{12} \approx \sqrt{3.4}.

Now the natural question arises: if the round sphere is not optimum, then what is the optimum?

At this point I looked into the literature a little bit, turns out this problem is quite well-studied and there is a conjecture by Christopher Croke that the optimum is exactly \sqrt[4]{12}. (Of course this optimum is achieved by our singular triangle metric hence after smoothing it would be < \sqrt[4]{12}.

There is even some recent progress made by Alex Nabutovsky and Regina Rotman from (our!) University of Toronto! See this and this. In particular, one of the things they proved was that the shortest geodesic on a unit area sphere cannot be longer than 8, which I believe is the best known bound to date. (i.e. there is still some room to \sqrt[4]{12}.)

Random remark: The essential difference between this and the systolic questions is that the sphere is simply connected. So the usual starting point, namely ‘lift to universal cover’ for attacking systolic questions does not work. There is also the essential difference where, for example, the question I addressed above regarding whether the shortest geodesic is simple would not exist in systolic situation since we can always split the curve into two pieces and tighten them up, at least one would still be homotopically non-trivial. In conclusion since this question sees no topology but only the geometry of the metric, I find it interesting in its own way.

Intergal geometry and the minimax inequality in a nutshell

Conan1 Comment

The goal for most of the posts in this blog has been to take out some very simple parts of certain papers/subjects and “blow them up” to a point where anybody (myself included) can understand. Ideally the simple parts should give some inspirations and ideas towards the more general subject. This one is on the same vein. This one is based on parts of professor Guth’s minimax paper.

In an earlier post, we talked about the extremal length where one is able to bound the “largest possible minimum length” (i.e. the “maximum minimum length“) of a family of rectifiable curves under conformal transformation. When combined with the uniformization theorem in for surfaces, this becomes a powerful tool for understanding arbitrary Riemannian metrics (and for conformal classes of metrics in higher dimensions).

However, in ‘real life’ we often find what we really want to bound is, instead, the “minimum maximum length” of a family of curves, for example:

Question: Let \mathbb{D} \subseteq \mathbb{R}^2 be the unit disc. Given any family \mathcal{F} of arcs with endpoints on \partial \ \mathbb{D} and \mathcal{F} foliates \mathbb{D}, then how short can the logest arc in \mathcal{F} possibly be?

In other words, let \mathbb{F} be the collection of all possible such foliations \mathcal{F} as above, what is

\displaystyle \inf_{\mathcal{F} \in \mathbb{F}} \ \sup_{A \in \mathcal{F}} \ \ell(A)?

After playing around a little bit with those foliations, we should expect one of the fibres to be at least as long as the diameter ( i.e. no foliation has smaller maximum length leaf than foliating by straight lines ). Hence we should have

\displaystyle \inf_{\mathcal{F} \in \mathbb{F}} \ \sup_{A \in \mathcal{F}} \ \ell(A) = 2.

This is indeed easy to prove:

Proof: Consider the map f: S^1 \rightarrow S^1 where S^1 = \partial \ \mathbb{D}, f switches the end-points of each arc in \mathcal{F}. It is easy to check that f is a continuous, orientation reversing homeomorphism of the circle (conjugate to a reflection). Let p, q be its fixed points, L_1, L_2 be the two arcs in S^1 connecting p to q.

Let

g: z \mapsto -z

be the antipodal map on S^1.

Suppose p \neq g(q) then one of L_1, L_2 is longer than \pi, say it’s L_1.

Then we have

f \circ g (L_1) \subseteq L_1.

Hence f \circ g has a fixed point m in L_1, i.e. f(m) = -m.

There is a fibre A in \mathcal{F} with endpoints m, -m, the fibre must have length

\ell(A) \geq d(-m,m) = 2.

The remaining case is trivial: if p = g(q) then both L_1 and L_2 gets mapped into themselves orientation-reversingly, hence fixed points still exists.

Establishes the claim.

Instead of the disc, we may look at circles that sweep out the sphere (hence to avoid the end-point complications):

Theorem: Any one-parameter family of circles that foliates S^2 (except two points) must have the largest circle being longer than the equator.

This is merely applying the same argument, i.e. one of the circles needs to contain a pair of antipodal points hence must be longer than the equator.

In order for easier generalization to higher dimensions, with slight modifications, this can be formulated as:

Theorem: For any f: T^2 \rightarrow S^2 having non-zero degree, there is \theta \in S^1 where \ell(f(S^1 \times \{ \theta \}) is larger than the equator.

Hence in higher dimensions we can try to prove the same statement for largest image of a lever k-sphere under f: S^k \times S^{n-k} \rightarrow S^n. However before we do that I would like to highlight some intergal geometry machineries that are new to me but seemingly constantly used in proving those kinds of estimates. We shall get some idea of the method by showing:

Theorem: Let \mathbb{R}P^n be equipped with the round metric. p^k \subseteq \mathbb{R}P^n be a ‘flat’ k-dimensional plane. Then any k-chain z^k \subseteq \mathbb{R}P^n in the same k dimensional homology class as p^k must have volume at least as large as p^k.

Proof: Let Gr(\mathbb{R}P^n, n-k) be the set of all (n-k)-planes in \mathbb{R}P^n (i.e. the Grassmannian).

There is a standard way to associate a measure \mu on Gr(\mathbb{R}P^n, n-k):

Let \lambda be the Haar measure on SO(n+1), fix some Q \in Gr(\mathbb{R}P^n, n-k).

Since SO(n+1) acts on \mathbb{R}P^n, for open set S \subseteq Gr(\mathbb{R}P^n, n-k), we set

\mu(S) = \lambda( \{ T \in SO(n+1) \ | \ T(Q) \in S \}).

–The measure of a collection of planes is the measure of linear transformations that takes the given plane to an element of the set.

Now we are able to integrate over all (n-k)-planes!

For almost all Q \in Gr(\mathbb{R}P^n, n-k), since P is k-plane, we have | Q \cap P | = 1. ( not 1 only when they are ‘parallel’ )

Since [z] = [p] in H_k(\mathbb{R}P^n, \mathbb{Z}_2), for almost all Q, z intersects Q at least as much as P does. We conclude that for almost all Q, \ | z \cap Q | \geq 1.

Fact: There exists constant C such that for any k-chain \Sigma^k \in \mathbb{R}P^N,

\mbox{Vol}_k(\Sigma^k) = \mathbb{E}(|\Sigma \cap Q |).

The fact is obtained by diving the chain into fine cubes, observe that both volume and expectation are additive and translation invariant. Therefore we only need to show this for infinitesimal cubes (or balls) near 0. We won’t work out the details here.

Hence in our case, since for almost all Q we have | z \cap Q | \geq 1, the expectation \mathbb{E}(|z \cap Q |) \geq 1.

We therefore deduce

\mbox{Vol}_k(z) = \mathbb{E}(|z \cap Q |) \geq 1.

Establishes the theorem.

Remark: I found this intergal geometry method used here being very handy: in the old days I always try to give lower bounds on volume of stuff by intersecting it with planes and then pretend the ‘stuff’ were orthogonal to the plane, which is the worst case in terms of having small volume. An example of such bound can be found in the knot distorsion post where in order to lower bound the length we look at its intersection number with a family of parallel planes and then integrate the intersection.

This is like looking from one particular direction and record how many times did a curve go through each height, of course one would never get the exact length if we know the curve already. What if we are allowed to look from all directions?

I always wondered if we know the intersection number with not only a set of parallel planes but planes in all directions, then are there anything we can do to better bound the volume? Here I found the perfect answer to my question: by integrating over the Grassmannian, we are able to get the exact volume from how much it intersect each plane!

We get some systolic estimates as direct corollaries of the above theorem, for example:

Corollary: \mbox{Sys}_1(\mathbb{R}P^2) = \sqrt{\pi/2} where \mathbb{R}P^2 carries the round metric with total volume 1.

Back to our minimax problems, we state the higher dimensional version:

Wish: For any C^1 map f: S^k \times S^{n-k} \rightarrow S^n where S^n carries the standard round metric, there exists some \theta \in S^{n-k} with

\mbox{Vol}_k(f(S^k\times \{\theta\})) \geq \mbox{Vol}_k(E^k)

where E^k \subseteq S^n is the k-dimensional equator.

But what we have is that there is a (small) positive constant c(n,k) s.t. \mbox{deg}(f) \neq 0 implies

\displaystyle \sup_{\theta \in S^{n-k}} \mbox{Vol}_k(f(S^k \times \{\theta\})) \geq c(n,k) \mbox{Vol}_k(E^k)

(shown by an inductive application of the isomperimetric inequality on S^N, which is obtained from applying intergal geometry methods)

Extremal length and conformal geometry

Conan2 Comments

There has been a couple of interesting talks recently here at Princeton. Somehow the term ‘extremal length’ came up in all of them. Due to my vast ignorance, I knew nothing about this before, but it sounded cool (and even somewhat systolic); hence I looked a little bit into that and would like to say a few words about it here.

One can find a rigorous exposition on extremal length in the book Quasiconformal mappings in the plane.

Let \Omega be a simply connected Jordan domain in \mathbb{C}. f: \Omega \rightarrow \mathbb{R}^+ is a conformal factor on \Omega. Recall from my last post, f is a Lebesgue measurable function inducing a metric on \Omega where

\mbox{Vol}_f(U) = \int_U f^2 dx dy

and for any \gamma: I \rightarrow \Omega (I \subseteq \mathbb{R} is an interval) with ||\gamma'(t)|| = \bar{1}, we have the length of \gamma:

l_f(\gamma) = \int_I f dt.

Call this metric g_f on \Omega and denote metric space (\Omega, g_f).

Given any set \Gamma of rectifiable curves in U (possibly with endpoints on \partial U), each comes with a unit speed parametrization. Consider the “f-width” of the set \Gamma:

\displaystyle w_f(\Gamma) = \inf_{\gamma \in \Gamma} l_f(\gamma).

Let \mathcal{F} be the set of conformal factors f with L^2 norm 1 (i.e. having the total volume of \Omega normalized to 1).

Definition: The extremal length of \Gamma is given by

\mbox{EL}(\Gamma) = \displaystyle \sup_{f \in \mathcal{F}} w_f(\Gamma)^2

Remark: In fact I think it would be more natural to just use w_f(\Gamma) instead of w_f(\Gamma)^2 since it’s called a “length”…but since the standard notion is to sup over all f, not necessarily normalized, and having the f-width squared divide by the volume of \Omega, I can’t use conflicting notation. One should note that in our case it’s just the square of sup of width.

Definition:The metric (\Omega, g_f) where this extremal is achieved is called an extremal metric for the family \Gamma.

The most important fact about extremal length (also what makes it an interesting quantity to study) is that it’s a conformal invariant:

Theorem: Given h: \Omega' \rightarrow \Omega bi-holomorphic, then for any set of normalized curves \Gamma in \Omega, we can define \Gamma' = \{ h^{-1}\circ \gamma \ | \ \gamma \in \Gamma \} after renormalizing curves in \Gamma' we have:

\mbox{EL}(\Gamma) = \mbox{EL}(\Gamma')

Sketch of a proof: (For simplicity we assume all curves in \Gamma' are rectifiable, which is not always the case i.e. for bad maps h the length might blow up when the curve approach \partial \Omega' this case should be treated with more care)

This is indeed not hard to see, first we note that for any f: \Omega \rightarrow \mathbb{R}^+ we can define f' : \Omega' \rightarrow \mathbb{R}^+ by having

f^\ast (z) = |h'(z)| (f \circ h) (z)

It’s easy to see that \mbox{Vol}_{f^\ast}(\Omega') = \mbox{Vol}_{f}(\Omega) (merely change of variables).

In the same way, l_{f^\ast}(h^{-1}\circ \gamma) = l_f(\gamma) for any rectifiable curve.

Hence we have

w_{f^\ast}(\Gamma') = w_f(\Gamma).

On the other hand, we know that \varphi: f \mapsto f^\ast is a bijection from \mathcal{F}_\Omega to \mathcal{F}_{\Omega'}, deducing

\mbox{EL}(\Gamma) = \displaystyle ( \sup_{f \in \mathcal{F}} w_f(\Gamma))^2 = \displaystyle ( \sup_{f' \in \mathcal{F}'} w_{f'}(\Gamma'))^2 = \mbox{EL}(\Gamma')

Establishes the claim.

One might wonder how on earth should this be applied, i.e. what kind of \Gamma are useful to consider. Here we emphasis on the simple case where \Omega is a rectangle (Of course I would first look at this case because of the unresolved issues from the last post :-P ):

Theorem: Let R = (0,w) \times (0, 1/w), \Gamma be the set of all curves starting at a point in the left edge \{0\} \times [0, 1/w], ending on \{1\} \times [0, 1/w] with finite length. Then \mbox{EL}(\Gamma) = w^2 and the Euclidean metric f = \bar{1} is an extremal metric.

Sketch of the proof: It suffice to show that any metric g_f with \mbox{Vol}_f(R) = 1 has at least one horizontal line segment \gamma_y = [0,w] \times \{y\} with l_f(\gamma_y) \leq w. (Because if so, w_f(\Gamma) \leq w and we know w_{\bar{1}}(\Gamma) = w for the Euclidean length)

The average length of \gamma_y over y is

w \int_0^{1/w} l_f(\gamma_y) dy

= w \int_0^{1/w} (\int_0^w f(t, y) dt) dy = w \int_R f

By Cauchy-Schwartz this is less than w (\int_R f^2)^{1/2} |R|^{1/2} = w

Since the shortest curve cannot be longer than the average curve, we have w_f(\Gamma) \leq w.

Hence \mbox{EL}(\Gamma) = \displaystyle \sup_{f \in \mathcal{F}}w_f(\Gamma)^2 = w^2

Note it’s almost the same argument as in the proof of systolic inequality on the 2-torus.

Corollary: Rectangles with different eccentricity are not conformally equivalent (i.e. one cannot find a bi-homomorphic map between them sending each edge to an edge).

Remark: I was not aware of this a few days ago and somehow had the silly thought that there are conformal maps between any pair of rectangles while discussing with Guangbo >.< then tried to see what would those maps look like and was of course not able to do so. (there are obviously Riemann maps between the rectangles, but they don't send conners to conners, i.e. can't be extended to a conformal map on the closed rectangle).

An add-on: While I came across a paper of Odes Schramm, applying the techniques of extremal length, the following theorem seemed really cool.

Let G = (V, E) be a finite planar graph with vertex set V and edges E \subseteq V^2. For each vertex v we assign a simply connected domain D_v.

Theorem: We can scale and translate each D_v to D'_v so that \{ D_v \ | \ v \in V \} form a packing (i.e. are disjoint) and the contact graph of D'_v is G. (i.e. \overline{D'_{v_1}} \cap \overline{D'_{v_2}} \neq \phi iff (v_1, v_2) \in E.

Note: This is vastly stronger than producing a circle packing with prescribed structure.

On length and volume

Conan2 Comments

About a year ago, I came up with an simple argument for the following simple theorem that appeared in a paper of professor Guth’s:

Theorem: If U is an open set in the plane with area 1, then there is a continuous function f from U to the reals, so that each level set of f has length at most 10.

Recently a question of somewhat similar spirit came up in a talk of his:

Question: Let \langle \mathbb{T}^2, g \rangle be a Riemannian metric on the torus with total volume 1, does there always exist a function f: \mathbb{T}^2 \rightarrow \mathbb{R} s.t. each level set of f has length at most 10?

I have some rough thoughts about how might a similar argument on the torus look like, hence I guess it would be a good idea to review and (somewhat carefully) write down the original argument. Since our final goal now is to see how things work on a torus (or other manifolds), here I would only present the less tedious version where U is bounded and all boundary components of U are smooth Jordan curves. Here it goes:

Proof: Note that if a projection of U in any direction has length (one-dimensional measure) \leq 10, then by taking f to be the projection in the orthogonal direction, all level sets are straight with length \leq 10 (see image below).

Hence we can assume any 1-dimensional projection of U has length \geq 10. A typically bad set would ‘span’ a long range in all directions with small area, it can contain ‘holes’ and being not connected:

Project U onto x and y-axis, by translating U, we assume \inf \pi_x(U) = \inf \pi_y(U) = 0. Look at the measure 1 set S in the middle of \pi_y(U) (i.e. a measure 1 set [a,b] \cap \pi_y(U) with the property m_1(\pi_y(U) \cap [0,a]) = m_1(\pi_y(U) \cap [b, \infty])

By Fubini, since the volume \pi_y^{-1}(S) is at most 1, there must be a point p\in S with m_1(\pi_y^{-1}(p))\leq 1:

Since the boundary of U is smooth, we may find a very small neighborhood B_\delta(p) \subseteq \mathbb{R} where for each q \in B_\delta(p), m_1(\pi_y^{-1}(q) \leq 1+\epsilon. (we will call this pink region a ‘neck’ of the set for it has small width and is roughly in the middle)

Now we define a \varphi_1: U \rightarrow \mathbb{R}^2 that straches the neck to fit in a long thin tube (note that in general \pi_y(U) may not be connected, but everything is still well-defined and the argument does go through.) and then bend the neck to make the top chunk vertically disjoint from the bottom chunk.

We can take \varphi so that \varphi^{-1} sends the vertical foliation of \varphi(U) to the following foliation in U (note that here we drew the neck wider for easier viewing, in fact the horizontal lines are VERY dense in the neck).

If the y-projection of the top or bottom chunk is larger than 2, we repeat the above process t the chunks. i.e. Finding a neck in the middle measure 1 set in the chunk, starch the neck and shift the top chunk, this process is guaranteed to terminate in at most m_1(\pi_y(U)) steps. The final \varphi sends U to something like:

Where each chunk has y-width L between 1 and 2.

Define f = \pi_x \circ \phi.

Claim: For any c \in \mathbb{R}, m_1(f^{-1}(c)) \leq 5.

The vertical line x=c intersects \varphi(U) in at most one chunk and two necks, taking \varphi^{-1} of the intersection, this is a PL curve C with one vertical segment and two horizontal segment in U:

The total length of f^{-1}(c) = C \cap U is less than 2+2\delta (length of U on the vertical segment) + 2 \times (1+\epsilon) (length of U on each horizontal segment). Pick \epsilon, \delta both less than 1/4, we conclude m_1(f^{-1}(c)) < 5.

Establishes the theorem.

Remark:More generally,any open set of volume V has such function with fibers having length \leq 5 \sqrt{V}. T he argument generalizes by looking at the middle set length \sqrt{V} set of each chunk.

Moving to the torus

Now let’s look at the problem on \langle \mathbb{T}^2, g \rangle, by the uniformization theorem we have a flat torus T^2 = \mathbb{R}^2/\Gamma where \Gamma is a lattice, \mbox{vol}(T^2) = 1 and a function h: T^2 \rightarrow \mathbb{R}^{+} s.t. \langle T^2, h g_0 \rangle is isometric to \langle \mathbb{T}^2, g \rangle. g_0 is the flat metric. Hence we only need to find a map on T^2 with short fibers.

Note that

\int_{T^2} h^2 d V_{g_0} = 1

and the length of the curve \gamma from p to q in \langle T^2, h \dot{g_0} \rangle is

\int_I h |\gamma'(t)| dt.

 

Consider T^2 as the parallelogram given by \Gamma with sides identified. w.l.o.g. assume one side is parallel to the x-axis. Let L be a linear transformation preserving the horizontal foliation and sends the parallelogram to a rectangle.

Let F be a piece-wise isometry that “folds” the rectangle:

(note that F is four-to-one except for on the edges and the two medians)
Since all corresponding edges are identified, $lates F$ is continuous not only on the rectangle but on the rectangular torus.

Now we consider F \circ L, pre-image of typical horizontal and vertical lines in the small rectangle are union of two parallel loops:

Note that vertical loops might be very long in the flat T^2 due to the shear while the horizontal is always the width.

(to be continued)