Spheres of small diameter with long sweep-outs

We prove the absence of a universal diameter bound on lengths of curves in a sweep-out of a Riemannian 2-sphere. If such bound existed it would yield a simple proof of existence of short geodesic segments and closed geodesics on a sphere of small diameter.


Introduction
By a sweep-out of a Riemannian 2-sphere M = (S 2 , g) we mean a non-contractible loop γ t in (ΛM, Λ 0 M ), where ΛM denotes the space of free loops on M and Λ 0 M denotes the space of constant loops. In other words, γ t is a 1-parameter family of closed curves, starting and ending at a point and inducing a non-zero degree map f : S 2 → M . If there exists a sweep-out by loops of length ≤ L then the standard minimax argument implies that there exists a non-trivial closed geodesic on M of length ≤ L ( [B]).
Moreover, if there exits a sweep-out of M by loops with 2 fixed points and of length ≤ L then for any two points a, b ∈ M there exists k distinct geodesic segments from a to b of length ≤ 2kL + 2diam(M ) (cf. [NR1]). If we could take L ≤ Cdiam(M ) we would obtain a linear in k bound on the length of k-th shortest geodesic segment in terms of the diameter.
Yet A.Nabutovsky and R.Rotman observe (see [N], [NR1], [NR2]) that example of S.Frankel and M.Katz [FK] (see below) suggests that for any C there exists a sphere with no sweep-out obeying this inequality. The author learnt from Regina Rotman that a similar conjecture was independently made by S.Sabourau. In this note we prove this conjecture.

Main result
Let γ : I × S 1 → M be a 1-parameter family of free loops (we write γ t (s) for γ(t, s)), such that γ 0 (s) and γ 1 (s) are constant loops. γ t induces a map f : S 2 → M , such that γ(t, s) = f • p(t, s), where p : I × S 1 → S 2 is the suspension map that collapses {0} × S 1 to the South pole and {1} × S 1 to the North pole of S 2 . If deg(f ) = 0 we call γ t a sweep-out.
Theorem 1. For any C > 0 there exists a Riemannian 2-sphere M of diameter ≤ 1, such that for any sweep-out γ t of M there is a loop γ t 0 of length ≥ C.

F.Balacheff and S.Sabourau in
where (γ t ) runs over the families of loops inducing f : S 2 → M of degree ±1. In [S,Remark 4.10] S.Sabourau constructs a sequence M n of Reimannain two-spheres such that Theorem 1 implies the analogous result with daim(M n ) in place of Area(M n ).
F.Balacheff and S.Sabourau prove ( [BS]) that if 1-parameter families of loops in the definition of the diastole are replaced with 1-parameter families of one-cycles then every Riemannian 2-sphere satisfies dias(M ) ≤ C Area(M ) for a universal constant C.
Proof. We use the example of S.Frankel and M.Katz [FK]. For any natural number N they embed a binary tree T of height N in a 2-dimensional disc D and define a Riemannian metric on D, such that the distance between any two non-adjacent edges of T dist(e i , e j ) ≥ 1/2, but the diameter diam(D) ≤ 1. They prove that for every homotopy of closed curves γ t with γ 0 = ∂D and γ 1 = { * } there is an intermediate curve γ t 0 that intersects at least O(N/logN ) edges of T and hence must be at least O(N/logN ) long.
Let M = (S 2 , g) be a sphere of diameter less than 1 containing the disc of Frankel and Katz with an embedded binary tree T of height N . Consider a sweep-out γ t and let f : S 2 → M be the induced map from the suspension of S 1 to M (γ t (s) = f • p(t, s)).
Suppose that We have the following commutative diagram ,v denote the restriction of f t 0 ,v to the south sphere S t 0 ∪{y} and f N t 0 ,v denote the restriction to the north sphere N t 0 ∪ {y}. Observe that from the induced commutative diagram for the second homology groups we have deg . Indeed, the map q 2♯ : H 2 (S 2 ) → H 2 (S 2 ∨ S 2 ) sends a generator 1 to an element (1, 1) ∈ Z × Z and ( Let A ⊂ [0, 1] denote the set of all t such that gamma t does not pass through v. We define a function d v : We need two simple facts about d v (t): ) is empty and define a quotient map q collapsing M \ D to a point. For each t ∈ [t 1 , t 2 ] we can define a map between spheres Let v 1 , v 2 be two vertices of T connected by an edge e. If γ t does not intersect e then d v 1 (t) = d v 2 (t) Proof: Since γ t does not intersect e we have that v 1 and v 2 belong to the same connected Without loss of generality we may assume that the images of North and South poles under f are not vertices of T . Now we observe that d v (0) = deg(f ) = 0 and d v (1) = 0 for all vertices v.
The rest of the proof proceeds as in [FK]. Let V be the set of vertices of T and K(t) = #{v ∈ V : d v = 0}. We may perturb the homotopy slightly, so that γ t passes through no more than one vertex for each t. Observation 1 implies that as t varies between 0 and 1 K(t) will attain every value between 0 and 2 N − 1 (recall that N is the height of T ).
Consider what values K(t) can attain if γ t intersects only one edge of T . Let v 1 and v 2 be two vertices of an edge e at distances (in the standard metric on the tree) i and i + 1, respectively, from the root. By Observation 2 we have the following possibilities for the value of K(t): the number of vertices in the connected component of M \ γ t that contains v 1 (2 N − 2 N −i−1 ), the number of vertices in the connected component that contains v 2 (2 N −i−1 − 1) or one of the min or max values (0 and 2 N − 1). Since this is true for every d i K(t) can attain at most 2(N − 1) + 2 possible different values if γ t intersects only one edge of T . Similarly, if have exactly j intersections then there are at most (2N ) j choices for the value of K(t). If throughout the homotopy γ t intersects at most k edges then K(t) attains no more than k j=1 (2N ) j ≤ (2N ) k+1 distinct values. Since all possible values are attained, we have (2N ) k+1 ≥ 2 N − 1 and hence k ≥ O(N/logN ).
Since the distance between any two non-adjacent edges is greater than 1/2 we have that for some t γ t will be at least O(N/logN ) long.