A ZOLL COUNTEREXAMPLE TO A GEODESIC LENGTH CONJECTURE

FLORENT BALACHEFF^{1}, CHRISTOPHER CROKE^{2}, AND MIKHAIL G. KATZ^{3}

Abstract. We construct a counterexample to a conjectured inequality L ≤ 2D, relating the diameter D and the least length L of a nontrivial closed geodesic, for a Riemannian metric on the 2sphere. The construction relies on Guillemin’s theorem concerning the existence of Zoll surfaces integrating an arbitrary infinitesimal odd deformation of the round metric. Thus the round metric is not optimal for the ratio L/D.

Contents

1. Zoll surfaces and Guillemin deformation 1 2. Amply negative odd functions 3 3. Diameter of Guillemin deformation 4 4. Fine sets and their properties 6 5. Existence of amply negative functions 7 References 10

1. Zoll surfaces and Guillemin deformation

Given a Riemannian metric on the 2-sphere, we consider its diameter D and the length L of its shortest nontrivial closed geodesic. The first inequality relating the two invariants was obtained by the secondmentioned author [Cr88], who proved the bound L ≤ 9D. The constant in the inequality was successively improved by M. Maeda [Ma94], A. Nabutovsky and R. Rotman [NR02], and S. Sabourau [Sa04]. The best known bound is L ≤ 4D. Nabutovsky and Rotman conjectured the inequality L ≤ 2D [NR02, Introduction], meaning that the round

Date: November 13, 2007. 1991 Mathematics Subject Classification. 53C23, 53C22 . Key words and phrases. Closed geodesic, diameter, Guillemin deformation,

sphere, systole, Zoll surface.

1Supported by the Swiss National Science Foundation. 2Supported by NSF grants DMS 02-02536 and DMS 07-04145. 3Supported by the Israel Science Foundation (grants 84/03 and 1294/06).

metric of S^{2} is optimal for the relationship between these two invariants. We give a few examples of surfaces satisfying the case of equality L = 2D:

(1) a surface of revolution in ℝ^{3} obtained from an ellipse with major axis on the x-axis;

(2) a circular “pillow”, obtained by doubling the flat unit disk;

(3) a more general pillow obtained by doubling the region enclosed by a closed curve of constant width in the plane;

(4) rotationally invariant Zoll surfaces.

The existence of such diverse examples may have led one to expect that none of these metrics are optimal for the ratio L/D.

It turns out that a counterexample to the inequality L ≤ 2D may be found among Zoll surfaces, namely surfaces all of whose geodesics are closed, and whose prime geodesics all have equal length 2π. More precisely, while the rotationally symmetric Zoll surfaces do satisfy (the boundary case of equality of) the conjectured inequality, there exist other families of Zoll surfaces such that L > 2D. Such surfaces can be obtained as smooth variations of the round metric.

Let (S^{2}, g_{0}) be the 2-sphere endowed with the round metric. Denote by a : S^{2} → S^{2} its antipodal map. Let C_{odd}^{∞}(S^{2}, ℝ) be the space of smooth odd functions on S^{2}, i.e. smooth real valued functions f satisfying f ∘a = −f. The following existence theorem for Zoll surfaces is due to V. Guillemin [Gu76].

Theorem 1.1 (Guillemin). For every f ∈ C_{odd}^{∞}(S^{2}, ℝ), there exists a smooth one-parameter family g_{t} = Ψ_{t}^{f}g_{0} of smooth Zoll metrics such that Ψ_{0}^{f} = 1, the conformal factor Ψ_{t}^{f} satisfies (dΨ_{t}^{f}/dt)|_{t=0} = f, and all prime periodic geodesics of (S^{2}, g_{t}) have length 2π.

Note that this result is a converse to P. Funk’s theorem [Fu13], to the effect that a smooth variation g_{t} = Φ_{t}g_{0} of the round metric by smooth Zoll metrics necessarily satisfies (dΦ_{t}/dt)|_{t=0} ∈ C_{odd}^{∞}(S^{2}, ℝ). A survey of Zoll surfaces appeared in [Bes78, Chapter 4], see also [LM02].

We exploit such Guillemin deformations to show that the round metric is not even a local maxinum of the ratio L/D among Zoll surfaces. The precise statement of our result relies on the notion of a Y-like set.

Definition 1.2. A subset of the unit circle is called Y-like if it contains a triple of vectors {u, v, w} such that there exist positive real numbers a > 0, b > 0, c > 0 satisfying au + bv + cw = 0. A subset of the unit tangent bundle US^{2} of S^{2} will be called Y-like if its intersection with the unit tangent vectors at p is Y-like for every p ∈ S^{2}.

Note that a subset of the unit circle is Y -like if and only if every open semicircle contains an element of the set.

We will denote by ds_{0} the element of length for the round metric g_{0} on the sphere. The notion of an amply negative function is motivated in Remark 2.4 below.

Definition 1.3. An odd function f is called amply negative if the set of unit tangent directions to great half-circles τ satisfying ∫_{τ} fds_{0} < 0, is a Y-like subset of US^{2}.

Theorem 1.4. If f is an amply negative function then the smooth variation {g_{t}} = Ψ_{t}^{f}g_{0} of the round metric g_{0} by smooth Zoll metrics satisfies L(g_{t}) > 2D(g_{t}) for sufficiently small t > 0.

Combined with the existence of amply negative functions proved in Section 5, our theorem yields the desired counterexample.

These metrics also provide a counterexample to another conjecture of Nabutovsky and Rotman [NR07, Conjecture 1, p. 13]. Their conjecture would imply that for every point p of a closed Riemannian manifold (M, g), there is a nontrivial geodesic loop at p of length at most 2D(g). Here a geodesic loop is a geodesic segment with identical endpoints. This conjecture is easily seen to be true for non-simply-connected manifolds, by exploiting non-contractible loops, cf. [Ka07]. In our examples, the shortest geodesic loop at every p has length 2π, while the diameter is strictly smaller than π.

Sections 2 and 3 contain a proof of Theorem 1.4 modulo on the existence of amply negative functions. The existence of the latter is verified in Sections 4 and 5.

2. Amply negative odd functions

Our goal is to find amply negative functions f ∈ C_{odd}^{∞}(S^{2}, ℝ), such that the corresponding Guillemin deformation g_{t} of the standard round metric g_{0} satisfies D(g_{t}) < π for t small enough (while all geodesics remain closed of length 2π). By the compactness of the unit tangent circle bundle US^{2}, we obtain the following lemma.

Lemma 2.1. For every amply negative function f there is a constant ν(f) > 0 with the following property. For every (p, v) ∈ US^{2}, there is a great half-circle τ issuing from p ∈ S^{2}, forming an acute angle with v, and satisfying ∫_{τ} fds_{0} < −ν(f).

Denote by L_{0} the length functional with respect to the round metric g_{0}. Given a geodesic segment γ of length L_{0}(γ) < π, we denote by P_{γ} the 1-parameter family of piecewise geodesic paths with the following two properties:

• the path joins the endpoints of γ,

• the path consists of a pair of imbedded geodesic segments of equal length.

Elements of P_{γ} are parametrized by the non-smooth midpoint of the piecewise geodesic path, which traces out the equidistant great circle of the two endpoints. We let SP_{γ} ⊂ P_{γ} be the closed subfamily consisting of the shorter paths, namely

SP_{γ} = {τ ∈ P_{γ}| L_{0}(τ) ≤ π} .

If γ is a great semi-circle, define P_{γ} to be the circular family of great half-circles joining the endpoints of γ, and the subfamily SP_{γ} to be the family of paths forming either an acute or a right angle with γ at the endpoints. The following lemma is obvious but crucial.

Lemma 2.2. The family SP_{γ} for a geodesic segment γ with L_{0}(γ) = π is the limit of the families SP_{γ_{i}} for subarcs γ_{i} of γ of length tending to π. In fact, if γ_{i} is any sequence of minimizing geodesic segments converging to γ, then SP_{γ_{i}} converges to SP_{γ}.

Our main technical tool in the next section will be the following result.

Lemma 2.3. If f is amply negative then there is an ϵ > 0 so that for all geodesic segments γ with π −ϵ ≤ L_{0}(γ) ≤ π, there is a path τ ∈ SP_{γ} with ∫_{τ} fds_{0} < −ν(f).

Proof. If no such ϵ exists, then there is a sequence {γ_{i}} with L_{0}(γ_{i}) < π and L_{0}(γ_{i}) → π such that all τ ∈ SP_{γ_{i}} satisfy ∫_{τ} fds_{0} ≥ −ν(f). There is a convergent subsequence such that γ′_{i}(0) → γ′(0) with L_{0}(γ) = π. By Lemma 2.2, the family SP_{γ_{i}} converges to SP_{γ}. By the continuity of f, for every τ ∈ SP_{γ} we have ∫_{τ} fds_{0} ≥ −ν(f), contradicting the assumption that the function f is amply negative . □

Remark 2.4. Given a piecewise geodesic τ over which the integral of f is negative, we will show in the next section that the length of τ decreases under the Guillemin deformantion. If, in addition, the curve τ has length at most π with respect to the metric g_{0}, then the length with respect to the metric g_{t} will be shorter than π. That is why we need to work with piecewise geodesics specifically in SP_{γ}. In order to make the continuity argument above work, one needs to find in each SP_{γ}, a curve τ over which f integrates negatively. This leads to the amply negative condition we introduced.

3. Diameter of Guillemin deformation

Let Ψ^{f} be the conformal factor of the Guillemin deformation, as in Theorem 1.1 above. Thus, the metric g_{t} = Ψ_{t}^{f}g_{0} is Zoll, while Ψ_{0}^{f} = 1 and (dΨ_{t}^{f}/dt)|_{t=0} = f. Consider the arclength parametrisation τ(s) of a path τ ⊂ S^{2} for the round metric g_{0}.

Lemma 3.1. The energy E_{t}(τ) of a path τ ⊂ S^{2} for the metric g_{t} satisfies

\frac{dE_{t}}{dt} |_{t=0} = ∫_{τ} f ∘ τ ds_{0}.

Proof. We have

\frac{d}{dt}E_{t}(τ) = \frac{d}{dt} ∫_{0}^{L_{0}(τ)} g_{t}(τ′(s), τ′(s))ds

= \frac{d}{dt} ∫_{τ} Ψ_{t}^{f} ∘ τ ds_{0}

= ∫_{τ} (\frac{d}{dt}Ψ_{t}^{f}) ∘ τ ds_{0}

= ∫_{τ} f ∘ τ ds_{0}

at t = 0. □

Proposition 3.2. If f is amply negative, then the associated Guillemin deformation g_{t} = Ψ_{t}^{f}g_{0} as in Theorem 1.1 satisfies D(g_{t}) < π for all sufficiently small t > 0.

Proof. Denote by L_{t} and d_{t} the length and the distance with respect to the metric g_{t}. Let ϵ > 0 be chosen as in Lemma 2.3, and let A_{ϵ} ⊂ S^{2}×S^{2} be the set of nearly antipodal pairs, defined by setting

A_{ϵ} = {(p, q) ∈ S^{2} × S^{2} |d_{0}(p, q) ≥ π − ϵ}

By continuity, there is a δ > 0 such that whenever 0 < t < δ, we have

(3.1)  d_{t}(p, q) < π for all (p, q) ∉ A_{ϵ}.

Now let (p, q) ∈ A_{ϵ}, and γ a minimizing geodesic joining them. Let

N(γ) = {τ ∈ SP_{γ} |∫_{τ} fds_{0} ≤ −ν(f)} ,

and let N = {τ ∈ N(γ)|π − ϵ ≤ L_{0}(γ) ≤ π}. By Lemma 2.3, whenever

π − ϵ ≤ L_{0}(γ) ≤ π,

the set N(γ) is non-empty. Furthermore, the sets N and N(γ) are compact. Now for small t > 0, define a continuous function F : N × ℝ → ℝ by setting F (τ, t) = \frac{dE_{t}(τ)}{dt}. By Lemma 3.1 and the definition of N, we have F (τ, 0) ≤ −ν(f). Hence by the compactness of N and the continuity of F there is a real δ′ > 0 so that for all 0 ≤ t ≤ δ′ and all τ ∈ N, we have F (τ, t) < −\frac{1}{2}ν(f). Therefore the energy given by the expression

∫_{0}^{L_{0}(τ)} g_{t}(τ′(s), τ′(s))ds_{0}

is strictly decreasing in t. Hence for 0 < t ≤ δ′, it is strictly smaller than the quantity

∫_{0}^{L_{0}(τ)} g_{0}(τ′(s), τ′(s))ds_{0} = L_{0}(τ).

In particular, we obtain for 0 < t ≤ δ′,

L_{t}(τ) = ∫_{0}^{L_{0}(τ)} \sqrt{g_{t}(τ′(s), τ′(s))}ds_{0}
≤ L_{0}(τ)^{\frac{1}{2}} ( ∫_{0}^{L_{0}(τ)} g_{t}(τ′(s), τ′(s))ds_{0})^{\frac{1}{2}}
< L_{0}(τ).

Thus for each pair (p, q) ∈ A_{ϵ} and every 0 < t ≤ δ′, there is a path τ from p to q with L_{t}(τ) < L_{0}(τ) ≤ π. Hence dist_{t}(p, q) < π. Combined with (3.1), this yields the diameter bound D(g_{t}) < π whenever 0 < t < min{δ, δ′}, proving the proposition as well as Theorem 1.4. □

4. Fine sets and their properties

Recall that an open hemisphere is an open ball of radius π/2 centered at any point of the unit sphere. The construction of amply negative functions in Section 5 exploits fine sets, in the following sense.

Definition 4.1. A spherical pointset X is called fine if the following three conditions are satisfied:

(1) no triple of X is collinear;

(2) no triple of great circles pp′, where p, p′ ∈ X, is concurrent other than at points of X (as well as their antipodal points);

(3) every open hemisphere contains at least 3 of the points of X.

Note that the non-collinearity implies, in particular, that X contains no pair of antipodal points. Meanwhile, condition (3) implies that at every point of the sphere, there is a Y-like set of tangent directions leading to points of X.

To see that fine sets exist, start with the set of 4 vertices of the regular inscribed tetrahedron. This gives a set with at least one point in every open hemisphere. We replace each point of the tetrahedron by a generic triple of nearby points. The non-collinearity and non-concurrency follow from genericity, and property (3) follows by construction.

Definition 4.2. Given a fine pointset X, choose ϵ(X) > 0 such that:

(1) the closed ϵ(X) balls centered at the points of X ∪ −X are disjoint;

(2) there are at least 3 points of X in B(p, π/2−ϵ(X)) for every p ∈ S^{2}.

We note that property (3) of fineness along with standard compactness arguments shows that such a positive ϵ(X) exists.

Lemma 4.3. Let X be a fine set and choose ϵ(X) as above. Let Σ the set of unit vectors in US^{2} tangent to geodesic segments τ of length π satisfying the following two conditions:

• τ(0, π) ∩ −X = ∅;

• τ(ϵ(X), π − ϵ(X)) ∩ X ≠ ∅.

Then Σ is Y-like.

Proof. Fix a unit vector v at p ∈ S^{2} and let γ be the corresponding geodesic segment of length π. We need to find a w ∈ Σ at p making an acute angle with v. Let H be the (closed) hemisphere obtained as the union of the τ ∈ SP_{γ}. Then by assumption there are at least three points of X (call them p_{1}, p_{2}, and p_{3}) in the interior of H and at a distance greater than ϵ(X) from the boundary of H hence the endpoints of γ. Hence there are at least 3 geodesic segment τ_{1}, τ_{2}, and τ_{3} in the interior of SP_{γ} passing through the points p_{i}, i = 1, 2, 3. If two of these paths coincide (say τ_{1} = τ_{2}) then τ_{1} passes through p_{1} and p_{2} so the initial point p of γ is not in X and τ_{1} avoids −X (by condition 1 of being fine). If all of these paths are pairwise distinct and also pass through points of −X (say −p_{4},−p_{5}, and −p_{6} respectively) then the initial point p of γ would lie on the 3 great circles p_{1}p_{4}, p_{2}p_{5}, p_{3}p_{6} which contradicts either condition 2 (if p ∉ X ∪ −X) or condition 1 (if p ∈ X ∪ −X). Thus we see that there is a τ in the interior of SP_{γ} containing an element of X at least ϵ(X) from the endpoints, and no element of −X in its interior. The tangent vector of τ is the w we seek. □

5. Existence of amply negative functions

The goal of this section is to prove the following proposition.

Proposition 5.1. There exist amply negative functions.

We will costruct such functions by defining odd functions that approximate the sum of ±δ (Dirac delta) functions centered at points of −X and X for a fine set X. For our approximate δ functions we take for each p ∈ S^{2} the smooth function δ_{p}^{ϵ} with support included in the ball B(p, ϵ) with

δ_{p}^{ϵ}(q) = exp(1/ϵ) · exp (\frac{1}{d(p, q) − ϵ})

for q ∈ B(p, ϵ).

We will use the following (nearly obvious) lemma.

Lemma 5.2. if γ is a diameter of B(p, ϵ) (i.e. geodesic through the center of length 2ϵ) and τ is any geodesic segment in B(p, ϵ) then ∫_{τ} δ_{p}^{ϵ} ≤ ∫_{γ} δ_{p}^{ϵ} with equality holding if and only if τ is also a diameter.

Proof. To see this (since δ_{p}^{ϵ} ≥ 0) we can assume (by extending τ if needed) that τ runs from a boundary point to a boundary point and has length 2l < 2ϵ. Since for t ≤ l we have d(p, τ(t)) ≥ ϵ−t = d(p, γ(t)) we have

∫_{τ} δ_{p}^{ϵ} = 2 ∫_{0}^{l} δ_{p}^{ϵ}(τ(t))dt ≤ 2 ∫_{0}^{l} δ_{p}^{ϵ}(γ(t)) < 2 ∫_{0}^{ϵ} δ_{p}^{ϵ}(γ(t)) = ∫_{γ} δ_{p}^{ϵ}.

□

We are now ready to define our functions.

Definition 5.3. For ϵ(X) > ϵ > 0 set

f_{X}^{ϵ} = ∑_{p_{i}∈X}(δ_{−p_{i}}^{ϵ} − δ_{p_{i}}^{ϵ})

Note that f_{X}^{ϵ} is a smooth odd function. We will now prove that for sufficiently small ϵ > 0 the function f_{X}^{ϵ} is amply negative .

Lemma 5.4. For every v ∈ US^{2} there is an ϵ(v) with ϵ(X) > ϵ(v) > 0 and an open neighborhood U(v) of v in US^{2} (note that the base point also varies) such that for all w ∈ U(v) there is a geodesic segment τ of length π whose initial tangent vector makes an acute angle with w (hence it starts at the base point of w) and

∫_{τ} f_{X}^{ϵ} < 0

for all ϵ(v) > ϵ > 0.

Note that the ϵ(v) > 0 we find in the proof below will tend to 0 as the base point of v tends to X (while not being in X). This turns out not to be a problem since by the compactness of US^{2} a finite number U(v_{i}) cover US^{2} and hence we can take any ϵ less than the smallest of the ϵ(v_{i}) and have that f_{X}^{ϵ} is amply negative. This therefore proves Proposition 5.1.

Proof of Lemma 5.4. To prove the lemma, we consider two cases. First assume that the base point of v is not in X∪−X. Let τ (whose existence is promised in Lemma 4.3) be a geodesic segment of length π making an acute angle with v that misses −X and passes through at least one p ∈ X that has distance greater than ϵ(X) from its endpoints. Thus we can choose ϵ(v) so small that τ misses B(q, 2ϵ(v)) for all q ∈ −X. Now for w in a small enough neighborhood U of v, let τ̄ be the geodesic segment of length π through the base point of w and p. For small enough U, τ̄ will still miss all the B(q, ϵ(v)) for q ∈ −X while τ̄ will make an acute angle with w. Thus, for ϵ(v) ≥ ϵ > 0, we have f_{X}^{ϵ} ≤ 0 along τ̄ and is negative near p so we see ∫τ̄ f_{X}^{ϵ} < 0.

In the second case the basepoint p_{0} of v is in X ∪ −X. We will assume p_{0} ∈ −X since the other case is the same (by reversing orientation of all geodesics). Note that any τ making an acute angle with v which intersects X in its interior cannot also intersect −X in its interior by property 1 of a fine set. So there are two (in fact three) geodesic segments τ_{1} and τ_{2} in the interior of SP_{γ} that pass through a p_{1} and a p_{2} respectively and no element of −X in its interior and such that p_{1} and p_{2} have distance greater than ϵ(X) from p_{0} and −p_{0}. Now we choose ϵ(v) > 0 so small that for all q ∈ −X and q ≠ p_{0}, B(q, 2ϵ(v)) miss both τ_{1} and τ_{2}. Again for w in a small neighborhood U of v let τ̄_{1} (resp τ̄_{2}) be the geodesic segment of length π starting at the basepoint of w and passing through p_{1} (resp. p_{2}).

If U is small enough we can assume both that τ_{i} make acute angles with w and that for all q ∈ −X with q ≠ p_{0}, τ̄_{1} and τ̄_{2} miss B(q, ϵ(v)). Along τ̄_{1} (resp. τ̄_{2}) we have, for ϵ(v) ≥ ϵ > 0, f_{X}^{ϵ} ≤ 0 except on

τ̄_{1} ∩ B(p_{0}, ϵ)

(respectively, on τ̄_{2} ∩ B(p_{0}, ϵ)). Both τ̄_{1} ∩ B(p_{0}, ϵ) and τ̄_{2} ∩ B(p_{0}, ϵ) cannot be diameters since that would put p_{0}, p_{1} and p_{2} on the same great circle (namely the one through p_{0} and the base point of w). So assume τ̄_{1} ∩ B(p_{0}, ϵ) is not a diameter. Then since we know on the other hand that τ̄_{1} ∩ B(p_{1}, ϵ) is a diameter, Lemma 5.4 tells us that ∫τ̄_{1} f_{X}^{ϵ} < 0. □

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Section de Mathématiques, Université de Genève, Suisse

  E-mail address: florent.balacheff@math.unige.ch

Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395 USA

  E-mail address: ccroke@math.upenn.edu

Mikhail G. Katz, Department of Mathematics, Bar Ilan University, Ramat Gan 52900 Israel

  E-mail address: katzmik@math.biu.ac.il