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Conformal Geometric Inequalities on the Klein Bottle

We prove three optimal conformal geometric inequalities of Blatter type on the Klein bottle. These inequalities provide conformal lower bounds of the volume and involve lengths of homotopy classes of curves that are candidates to realize the systole.


Introduction, preliminaries and results
Among all Riemannian metrics on a given compact differentiable manifold, the most interesting ones are those that extremize some Riemannian invariant. An interesting problem, for example, is to study metrics g which maximize the ratio (sys(M, g)) n /vol(M, g), where sys(M, g) is the systole of the n-dimensional Riemannian manifold (M, g). Nevertheless, as in many works related to this problem, various constraints can be put on the set of Riemannian metrics considered. For example we may restrict ourselves to the set of constant curvature metrics ( [1], [10], [7]) or to the set of metrics conformal to a given one ( [6], [3], [4], [5]).
As far as this work is concerned, we are interested in finding optimal conformal lower bounds of the volume of the Klein bottle (denoted by K) by (the product of) the least length of a class of non contractible curves. We will often consider (free) homotopy classes of curves that are candidates to realize the systole. Such inequalities were proved by Blatter on the Mobius band with boundary (denoted by M ) in 1961 ( [6]). Hoping to get a universal inequality on M , Blatter considers the family of curves F joining two points on the boundary and going through the central geodesic of the Mobius Band at least one time. If we denote by l * the least length of a curve that belongs to F and by l 1 the least length of a curve closed by the glide reflection generating π 1 (M ), Blatter (here we follow his notations) gets the following optimal conformal lower bound of the volume l * (g)l 1 (g) ≤ C β vol(g) where C β depends only on the conformal type of the metrics under consideration.
He was expecting to get an inequality of this type which holds for any Riemannian metric on the Mobius Band, but he found that this is not true. 1 We will show that a similar result holds for the Klein bottle (Corollary 1.2 below). Note that the family of curves F is a natural set of curves on the Klein bottle since it corresponds to the homotopy class of the vertical translation in π 1 (K) (see section 1.1 below).
1.1. The Klein bottle. It is the quotient of R 2 by the group generated by the maps σ : z →z + π and t : z → z + 2iβ. The flat metric induced by such a quotient will be denoted by g β . By the uniformization theorem, we know that every Riemannian metric g on the Klein bottle is conformally equivalent to a flat metric g β for some β ∈]0, +∞[. The latter is called the conformal type of the metric g.
The Klein bottle satisfies the isosystolic inequality The equality being achieved for a spherical metric outside a singular line (see [2], [13], [7]). For details and (many) open problems in systolic geometry see the book of Katz [11], the website http://u.math.biu.ac.il/∼katzmik/sgt.html, and the "deep" paper of Gromov [8].
We put the emphasis on the fact that the systole of any metric on the Klein bottle is realized by one of the following (free) homotopy class of curves: (1) The homotopy class of (a geodesic closed by) the glide reflection σ of π 1 (K). (2) The homotopy class of the vertical translation t of π 1 (K).
If g is a Riemannian metric on K, the least length of a curve in the first (resp. second, resp. third) class will be denoted by l σ (resp. l v , resp. l h ). Note that the extremal metric for the isosystolic inequality on the Klein bottle (K, g e ) verifies l σ (g e ) = l v (g e ) = π. Its conformal type is 2 ln(tan( 3π 8 )).

1.2.
The results. The first result of this paper studies optimal conformal inequalities of the form l v l σ ≤ C β vol (we call it "Geometric Inequality of Type σ − v"), where C β is a constant depending only on the conformal type β. It also describes the Riemannian metrics for which the best constant is achieved: and tan(b 0 ) = 2b 0 , then for every Riemannian metric g on the Klein bottle K conformal to g β we have vol(g).
Moreover, the equality is achieved by a spherical metric in the first case and a flat-spherical one in the second case.
Remark 1. The inequality in the first case was proved by Bavard in [5]. We complete here this study by obtaining an optimal inequality for the remaining conformal classes.

Corollary 1.2. There does not exist any (finite) constant c such that the inequality:
l v (g)l σ (g) ≤ c · vol(g) holds for every Riemannian metric g on the Klein bottle.
The second part of the paper is devoted to establishing optimal conformal inequalities of the form L σ l h ≤ C β vol, where L σ denotes the least length of a curve freely homotopic to a geodesic loop closed by an element of the subgroup < σ > generated by σ (this is the systole in the case of the Mobius Band with boundary). We call it "Geometric inequality of type σ n − v". Note that since the only candidates of curves for L σ in < σ > are σ and σ 2 , we have L σ = inf{l σ , l h }. Theorem 1.3. For every Riemannian metric g on the Klein bottle K conformal to g β , we have vol(g).
(2) If β > 2 ln 2 + √ 3 , then Moreover, the equality is achieved by a spherical metric in the first case and by a flat-spherical metric in the second case.
Since the supremum of the conformal constant C β over β is equal to 2 we have Corollary 1.4. For every Riemannian metric g on the Klein bottle K we have L σ (g)l v (g) ≤ 2vol(g) Though this inequality is optimal, the equality case is not achieved.
In the third part we establish optimal conformal inequalities of the form . This inequality has the advantage that the volume (power 3 2 to get a homogeneous invariant) is bounded by all three candidates of the systole: Theorem 1.5. For every Riemannian metric g on the Klein bottle K conformal to g β , we have the following inequality Where ω ∈]0, π 2 [ is given by Moreover, the equality is achieved by a flat-spherical metric.
3 , +∞[ is a continuous increasing onto function and then the supremum of C is +∞ as ω → 0 (i.e. when b → 0). Therefore we get the surprising corollary: Corollary 1.6. There does not exist any (finite) constant c such that the inequality l v (g)l σ (g)l h (g) ≤ c · vol(g) 3 2 holds for every Riemannian metric g on the Klein bottle K.

A maximality criterion
The proofs of our results are based on the method of extremal lengths initiated by Jenkins in [9], Gromov in [8], and Bavard in [4].
Let (M, g e ) be a compact Riemannian manifold, and let for each i ∈ {1, · · · , p}, S i be a set of rectifiable curves, such that S i ∩ S j = ∅ for all i = j. We denote by l i (g) the least length of a curve in S i with respect to a Riemannian metric g M conformal to g e .
For every Radon measure µ on Γ = S 1 ∪ · · · ∪ S p , we associate a measure * µ on M by setting for ϕ ∈ C 0 (M, R): where ϕ(γ) = ϕ • γ(s)ds, ds is the arc length of γ with respect to g e . Theorem 2.1. ( [3], [4] and [9]) The Riemannian metric g e is maximal in its conformal class with respect to the Riemannian ratio if there exists a positive measure µ on Γ such that m 1 l 1 (g e ) = m 2 l 2 (g e ) = · · · = m p l p (g e ) and * µ = dg e where, m i is the mass of the measure on S i , and dg e is the volume measure of (M, g e ).
Since our version of theorem 2.1 is not the same as in the references we gave, we give here a proof of it.
Proof. Let g be a Riemannian metric conformal to g e , (g = φg e ), we have Using the inequality of arithmetic and geometric means we get p m 1 · · · m p l 1 (g) · · · l p (g) 1 p ≤ m 1 l 1 (g) + · · · + m p l p (g) with equality if and only if m 1 l 1 (g) = m 2 l 2 (g) = · · · = m p l p (g) Finally, combining the two inequalities we get that, under the required conditions, the following inequality holds with equality if and only if g is proportional to g e .

Geometric Inequality of type σ − v on the Klein bottle
In the following, we prove geometric inequality of type σ − v. We use the notations introduced in section 1.
Proof of theorem 1.1. The inequality in the first case was proved by C.Bavard in [5] (corollary 2). For each β < 2 ln tan π 4 + b 0 2 the extremal metric is where f is a function invariant by the translation v → v + 2b and equal to cos in |v| ≤ b. Here we have β = 2 ln(tan( π 4 + b 2 )).
To prove the case when β > 2 ln tan π 4 + b 0 2 , we endow K with the metric G b : We aim to find ω (b 0 ≤ ω ≤ b) so that G b verifies the conditions of theorem 2.1.
We consider the two families of curves : where γ u (t) = (u, t) and |t| ≤ 2b.
(we used Abel integration equation, see [3] and [12] p. 66 for some details on the calculations).
For any b ≥ b 0 , the last equation has a unique solution ω ∈ [b 0 , π 2 [. Moreover, it can be checked easily that ω is related to β by 2 sin(ω) = β 2 − 2 ln tan π 4 + ω 2 cos 2 (ω) + 4ω cos(ω), Now, the volume of G b is equal to Finally, from inequality 2.1 and for ω solution of equation 3.1 we get the inequality which holds for every Riemannian metric g conformal to G b . The equality is achieved if and only if g is proportional to G b , and the result follows.
We denote by M be the Mobius band with boundary obtained by taking the quotient of R × [−β, β] by the group generated by the map σ : z →z + π. We also denote by g β the flat metric induced by such quotient.  Let b 0 be such that b 0 ∈]0, π 2 [ and tan(b 0 ) = 2b 0 , and let ω be such that b 0 ≤ ω < π 2 and sin(ω) = β − ln tan π 4 + ω 2 cos 2 (ω) + 2ω cos(ω). Then for every Riemannian metric g on the Mobius band M conformal to g β the following holds: vol(g).
(2) If β > ln tan π 4 + b 0 2 , then Moreover, the equality is achieved by a spherical metric in the first case and a flat-spherical one in the second case.

Geometric Inequality of Type σ n − v on the Klein bottle
In the following, we prove geometric inequality of type σ n − v. We use the notations introduced in section 1.
Proof of theorem 1.3. The inequality in the first case can be deduced from the proof of theorem 1.1 (first case) since in the case b < π . It only remains to notice that the conformal type is We deduce that for 0 < β ≤ 2 ln 2 + √ 3 , vol(g).
Note that for b > π 3 the horizontal lines closed by the translation σ 2 become shorter than the curves γ a θ and therefore L σ becomes achieved by such lines.
To prove the inequality in the second case, we endow K with the metric H b : We then consider the three families of curves : where δ a (t) = (t, a), |t| ≤ π/2, and We put on S 1 the measure µ 1 where, On S 2 , we put the measure µ 2 , where Finally, on S ′ 1 , we put the measure µ 3 , where The condition * µ 1 + * µ 2 + * µ 3 = dH b gives h(a) = sin(a) π cos(a) .
The mass m 1 of the measure on S 1 ∪ S ′ 1 is The mass of the measure on S 2 is m ′ and vol(H b ) = 2π We deduce from theorem 2.1 (first part) that which holds for every real number m ′ ∈ [0, π 2 ] and for every Riemannian metric g conformal to H b . Moreover, the equality sign is achieved for g = H b .
Next, we have L σ (H b ) = π, and l v (H b ) = 4b, and then the condition It follows that for every Riemannian metric g conformal to H b we have with equality if and only if g is proportional to H b . The conformal type of H b is Therefore when b ≥ π 3 , β ≥ 2 ln(2 + √ 3) and the desired result follows.  (2) If β > ln 2 + √ 3 , then Moreover, the equality is achieved by a spherical metric in the first case and a flat-spherical one in the second case.

Geometric Inequality of type σ − v − h on the Klein bottle
In the following, we prove geometric inequality of type σ − v − h. We use the notations introduced in section 1.
Proof of theorem 1.5. We define on K the metric where, ω is a positive real number such that ω ≤ b.
Finally we get from m ′ = π sin(ω) b+ω , together with the relation tan(ω) = b 2 −ω 2 2b−ω and theorem 2.1 that for all b > 0, we have vol(g) with equality if and only if g is proportional to E b .