Hyperelliptic surfaces are Loewner

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HYPERELLIPTIC SURF ACES ARE LOEWNER 1MIKHAIL G.KATZ ?AND ST ′EPHANE SABOURAU Abstract.We prove that C.Loewner’s inequality for the torus is satis?ed by conformal metrics on hyperelliptic surfaces X ,as well.In genus 2,we ?rst construct the Loewner loops on the (mildly singular)companion tori,locally isometric to X away from Weierstrass points.The loops are then transplanted to X ,and surgered to obtain a Loewner loop on X .In higher genus,we exploit M.Gromov’s area estimates for ε-regular metrics on X .Contents 1.Hermite constant and Loewner surfaces 12.Basic estimates 33.Hyperelliptic surfaces and ε-regularity 44.Proof of Theorem 3.1in genus two 5References 81.Hermite constant and Loewner surfaces The systole,sys π1(G ),of a compact non simply connected Riemann-ian manifold (X,G )is the least length of a noncontractible loop γ?X :sys π1(G )=min [γ]=0∈π1(X )length(γ).(1.1)This notion of systole is apparently unrelated to the systolic arrays of [Ku78].We will be concerned with comparing this Riemannian in-

variant to the total area of the metric,as in the Loewner inequality (1.3)for the torus.

The Hermite constant,denoted γn ,can be de?ned as the optimal constant in the inequality

sys π1(T n )2≤γn vol(T n )2/n ,(1.2)

2M.KATZ AND S.SABOURAU over the class of all ?at tori T n .Here γn is asymptotically linear in n ,cf.[LLS90,pp.334,337].The precise value is known for small n ,e.g.

one has γ

2=2

3,γ3=2

1

2

area(X ),(1.4)where the boundary case of equality in (1.4)is attained precisely when,on the one hand,the surface X is a real projective plane,and on the other,the metric is of constant Gaussian curvature.

We will prove the following result toward answering Question 1.2,cf.Theorem 3.1.

Theorem 1.3.Every metric on an orientable surface is Loewner if it lies in a hyperelliptic conformal class.In particular,every metric on the genus 2surface is Loewner.

HYPERELLIPTIC SURF ACES ARE LOEWNER13 While the precise value of the systolic ratio in genus2is unknown in the class of all metrics,an optimal systolic inequality for CAT(0) metrics does exist[KS05].The relevant literature and basic estimates are reviewed in Section2.We will state the main theorem in more detail in Section3and prove it for genus3or more.We will complete the proof in the genus2case in Section4.

2.Basic estimates

De?nition2.1.The systolic ratio of a metric G on a closed n-manifold is de?ned as

sysπ1(G)n

SR(G)=

4√

,as g→∞.

g

Another helpful estimate is found in[Gr83,Corollary5.2.B].Namely, every aspherical compact surface(Σ,G)admits a metric ball B= B p 12sysπ1(G),which satis?es

sysπ1(G)2≤4

sysπ1(G)satis?es the estimate

2

2r2

4M.KATZ AND S.SABOURAU

3.Hyperelliptic surfaces andε-regularity

Recall that a Riemann surface X is called hyperelliptic if it admits a degree2meromorphic function,cf.[Mi95,p.60-61]as well as[Mi95, Proposition4.11,p.92].The associated rami?ed double cover

Q:X→S2(3.1) over the sphere S2is conformal away from the2g+2rami?cation points, where g is the genus of X.Its deck transformation J:X→X is called the hyperelliptic involution.Such a holomorphic involution,if it exists, is uniquely characterized by the property of having precisely2g+2?xed points.The?xed points of J are called Weierstrass points.Their images under the map Q of(3.1)will be referred to as rami?cation points.

Theorem3.1.Let(X,G)be an orientable surface,where the metric G belongs to a hyperelliptic conformal class.Then(X,G)is Loewner. Since every genus2surface is hyperelliptic[FK92,Proposition III.7.2, page100],we obtain the following corollary.

Corollary3.2.Every metric on the genus2surface is Loewner. Note that this is the?rst improvement,known to the authors,on Gromov’s3/4bound(2.2)in over20years,for surfaces of genus be-low50,cf.Question1.2.No extremal metric has as yet been con-jectured in this genus,but it cannot be?at with conical singulari-ties[Sa04].The best available lower bound for the optimal systolic ratio in genus2can be found in[CK03,section2.2].

For genus g≥3,our Theorem3.1follows from Proposition3.6, cf.Remark2.2and[Kon03].Let us go over the de?nitions of[Gr83, 5.1].

De?nition3.3.Given a Riemannian metric G on X,the tension of a noncontractible loopγbased at x∈X is the upper bound of allδ>0 such that there exists a free homotopy ofγwhich diminishes the length ofγbyδ.The tension is denoted tens G(γ).

De?nition3.4.The height h G(x)of x∈X is de?ned as the lower bound of the tensions of the noncontractible loops based at x.Note that the height function is2-Lipschitz.

A metric G is said to beε-regular withε

Lemma3.5.Given any conformal metric G on(Σg,J)and a real num-berδ>0,there exists anε-regular,J-invariant,conformal metricˉG with a systolic ratio at least SR(G)?δ.

HYPERELLIPTIC SURF ACES ARE LOEWNER15 Proof.The proof appears in[Gr83,5.6.C′′]in the general case,and proceeds by a(?nite)sequence of modi?cations of the metric in suit-able small disks,while staying in the same conformal class.Note that averaging the metric by J improves the systolic ratio and does not change the conformal class.Thus we can assume that G is J-invariant. To adapt the proof to our situation,we perform the modi?cations in a J-invariant way. Proposition3.6.Every hyperelliptic surface(Σs,J)of genus g satis-?es the estimate

SR c(Σg)≤

4

2sysπ1(G).Thus,the disks of radius R=

1

8sysπ1(G)2?ε,

cf.(2.3).Therefore,we have

area(G)≥(2g+2) 1

6M.KATZ AND S.SABOURAU

De?nition4.2.A companion torus T(a,b,c,d)of X is a torus whose rami?cation locus{a,b,c,d}?S2is a subset of the rami?cation locus of X.

As in the proof of Proposition3.6,we can assume that the metric on X is invariant under J(see[BCIK05]).Therefore G descends to a metric G0,of half the area,on S2.Let’s choose four of the6rami?cation points,say a,b,c,d∈S2.Choose a double cover with rami?cation locus{a,b,c,d},denoted

T2(a,b,c,d)→S2.

Pulling back the metric G0to the torus T2(a,b,c,d),we obtain a metric of the same area as the surface X itself.This metric on the torus is smooth away from the two remaining points,where it has a conical singularity with total angleπaround each.Consider a Loewner loop

L L?T2(a,b,c,d)

on this torus,e.g.a systolic loop realizing(1.3).Let L be the projection of L L to S2.The simple loop L?S2separates the four points a,b,c,d into two pairs,say a,b on one side and c,d,on the other.If the lift of L to X closes up,we obtain a Loewner loop on X and the theorem is proved.Thus,we may assume that the following three equivalent conditions are satis?ed:

(1)the lift of L to X does not close up;

(2)the inverse image Q?1(L)?X under Q of(3.1)is connected;

(3)the loop L surrounds precisely3rami?cation points of Q. The last condition is equivalent to the?rst two since every based loop is homotopic,in the complement of the rami?cation points,to a composition of some loops from a?nite collection of“standard”simple based loops,circling each of the rami?cation points.Meanwhile,going once around such a standard loop clearly switches the two sheets of the cover.

De?nition 4.3.The simple loop L partitions the sphere into two hemispheres,H+and H?,with a,b,e∈H+and c,d,f∈H?where a,b,c,d,e,f are the6rami?cation points of Q.

Using a pair of companion tori,we will construct two loops on the sphere,de?ning two distinct partitions of the rami?cation locus into a pair of triples.The basic example to think of is the case of a centrally symmetric6-tuple of points,corresponding for instance to the curve

y2=x5?x,

HYPERELLIPTIC SURF ACES ARE LOEWNER17 and a pair of generic great circles,such that each of the four digons contains at least one rami?cation point.We now construct a companion torus T(a,b,e,f).

Consider a Loewner loop L L′?T2(a,b,e,f),and its projection L′?S2.If its lift to X closes up,the theorem is proved.Therefore assume that the lift of L′to X does not close up,i.e.L′surrounds exactly3 rami?cation points.Now L′separates the four points a,b,e,f into two pairs.Hence it de?nes a di?erent splitting of the six points into two triples.The connected components of L′∩H+form a nonempty?nite collection of disjoint nonsel?ntersecting arcsα.

Each arcαdivides H+into a pair of regions homeomorphic to disks. Such regions are partially ordered by inclusion.A minimal element for the partial order is necessarily a digon.Such a digon must contain at least one rami?cation point of Q(otherwise exchange the two sides of the digon between the loops L and L′,so as to decrease the total number of intersections,or else argue as in Lemma4.1).It is clear that there are at least two such digons in H+.

Hence one of them,denoted D?H+,must contain precisely one of the3rami?cation points of H+.We now exchange the two sides of D between the loops L,L′,obtaining two new loops M,M′.Each of the new loops surrounds a nonzero even number of rami?cation points. Since

length(M)+length(M′)=length(L)+length(L′),

one of the loops M or M′is no longer than Loewner.Moreover,its lift to X closes up,producing a Loewner loop on X,as required.

8M.KATZ AND S.SABOURAU

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Department of Mathematics and Statistics,Bar Ilan University, Ramat Gan52900Israel

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

Laboratoire de Math′e matiques et Physique Th′e orique,Universit′e de Tours,Parc de Grandmont,37400Tours,France

Mathematics and Computer Science Department,Saint-Joseph’s Uni-versity,5600City Avenue,Philadelphia,PA19131,USA E-mail address:sabourau@gargan.math.univ-tours.fr