Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 118, pp. 1–8.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu
GEODESICS OF QUADRATIC DIFFERENTIALS ON KLEIN SURFACES
MONICA ROS¸IU
Abstract. The objective of this article is to establish the existence of a local Euclidean metric associated with a quadratic differential on a Klein surface, and to describe the shortest curve in the neighborhood of a holomorphic point.
1. Introduction
In this paper we develop a technique, based on similar results for Riemann sur- faces, to determine the geodesics near holomorphic points of a quadratic differential on a Klein surface. Recent results about this topic are due to Andreian Cazacu [3], Bolo¸steanu [5], Bˆarz˘a [4] and Ro¸siu [7]. The complex double of a Klein surface is an important tool in connection with the study of the geometric structure near critical points of a quadratic differential on a Klein surface. Effectively, we define a metric associated with a quadratic differential on a Klein surface, which corresponds to a symmetric metric on its double cover. We introduce special parameters, in terms of which the quadratic differential has particularly simple representations and we give explicit descriptions of the geodesics near the holomorphic points of a quadratic differential.
Throughout this article, by a Klein surface we mean a Klein surface with an empty boundary, which is not a Riemann surface.
2. Preliminaries
Suppose thatX is a compact Klein surface, obtained from a compact surface by removing a finite number of points. We assume thatX has hyperbolic type. Our study is based on the following theorem (see [6]) due to Klein.
Theorem 2.1. Given a Klein surface X, its canonical (Riemann) double cover XC admits a fixed point free symmetry σ, such thatX is dianalytically equivalent with XC/hσi, where hσi is the group generated by σ. Conversely, given a pair (XC, σ) consisting of a Riemann surface X and a symmetry σ , the orbit space XC/hσiadmits a unique structure of Klein surface, such thatf :XC →XC/hσiis a morphism of Klein surfaces, provided that one regardsXC as a Klein surface.
2000Mathematics Subject Classification. 30F30, 30F35, 30F50.
Key words and phrases. Klein surface; meromorphic quadratic differential; geodesic.
c
2014 Texas State University - San Marcos.
Submitted February 2, 2014. Published April 22, 2014.
1
Following the next theorem we associate a surfaceN EC group with a compact Klein surfaceX. Details can be found in [7].
Theorem 2.2. Let X be a compact Klein surface of algebraic genusg ≥2. Then there exists a surfaceN EC groupΓ, such thatX andH/Γare isomorphic as Klein surfaces. Moreover, the complex double(XC, f, σ)is isomorphic withH/Γ+, where Γ+ is the canonical Fuchsian subgroup ofΓ.
Then Γ is the group of covering transformations of X and Γ+ is the group of conformal covering transformations ofX. Ifπ0 : H →H/Γ and π :H →H/Γ+ are the canonical projections, then we note with zb the local parameter near a pointPb ∈H, ez the local parameter nearPe ∈XC andz the local parameter near P =f(Pe)∈X.
Given A = {(Ui, φi) | i ∈ I} the dianalytic atlas on X, we define Ui0 = Ui× {i} × {1} andUi00=Ui× {i} × {−1}, i∈I. LetU ⊂Ui∩Uj 6=∅ be a connected component of Ui∩Uj. Then, we identify U × {i}× {δ} with U × {j}× {δ}, for δ=±1, ifφj◦φ−1i :φi(U)→Cis analytic andU× {i} × {δ}withU× {j} × {−δ}, forδ=±1, ifφj◦φ−1i :φi(U)→Cis antianalytic. As in Alling,XC is the quotient space ofX0=∪i∈IUi0∪ ∪i∈IUi00, (i, j)∈I×I, with all the identifications above.
We considerp:X0→XC the canonical projection andfUi=p(Ui0∪Ui00),i∈I.
Using Schwarz reflection principle, we can associate an analytic structure onXC byAe={(fUi,φei) :i∈I}, where
φei :fUi→C, φei(Pe) =
(φi(P), whenPe∈Ui0 φi(P), whenPe∈Ui00. Remark 2.3. Ae0={(Uei,φei) :i∈I}is also an analytic atlas onXC.
IfXC = (XC,A) ande XC= (XC,Ae0), thenXC isXC endowed with the second orientation.
Letf :XC→X,f(Pe) ={P , σ(e Pe)}be the covering projection, whereσ:XC→ XC is an antianalytic involution, without fixed points. If ez is a parameter near Pe∈XC, then ezis a parameter nearσ(Pe)∈XC.
Because two disjoint neighborhoodsUi0andUi00ofXC, lie over each neighborhood Ui ofX, we can make the restriction atUi0∪Ui00in the local study of the quadratic differentials onXC.
As in Alling and Greenleaf [2], we associate with the dianalytic atlas A onX, the nonzero (holomorphic) quadratic differential ϕ = (ϕi)i∈I on X, in the local parameters (zi)i∈I, such that on each connected component U of Ui ∩Uj, the following transformation law
ϕi(zi)(dzi)2=
(ϕj(zj)(dzj)2, whenφj◦φ−1i is analytic onφi(U) ϕj(zj)(dzj)2, whenφj◦φ−1i is antianalytic onφi(U) holds wheneverzi andzj are parameter values near the same point ofX.
The familyϕe= (ϕei)i∈Iof holomorphic function elements, in the local parameters (zei)i∈I,
ϕei(zei) =
(ϕi(zi), whenPe∈Ui0 ϕi(zi), whenPe∈Ui00,
(2.1)
where zei is a local parameter near P, is a (holomorphic) quadratic differential one XC, with respect to the analytic atlasAeonXC. By analogy, the familyϕe= (ϕei)i∈I of holomorphic function elements, in the local parameters (zei)i∈Iis a (holomorphic) quadratic differential onXC.
3. The natural parameter near holomorphic points
As a consequence of the above results, see [7], the order of a nonzero holomorphic quadratic differential ϕon X at a point P is a dianalytic invariant, thus it does make sense to define the zeroes of the quadratic differential ϕon a Klein surface.
A holomorphic point is either a regular point or a zero.
In this section, we extend the notion of natural parameter near a holomorphic point of a quadratic differential on a Riemann surface to a Klein surface, which is studied in [9].
LetP be a holomorphic point of the quadratic differentialϕand (Ui, zi), i∈I be a dianalytic chart atP. Then, we can takeUi to be sufficiently small so that a single valued branch of Φ(zi) =R p
ϕi(zi)dzi can be chosen. We introduce a local parameter
wi= Φ(zi) = Z
pϕi(zi)dzi
in Ui, uniquely up to a transformation wi → ±wi + const., such that on each connected componentU ofUi∩Uj, the following transformation law
wi=
(wj, whenφj◦φ−1i is analytic onφi(U) wj, whenφj◦φ−1i is antianalytic onφi(U)
holds wheneverzi andzj are parameter values near the same pointP ofX. By the definition of a quadratic differential on a Klein surface, we can see that the local parameterwi is locally well defined, up to conjugation, see [2].
Remark 3.1. The quadratic differential ϕ on the Klein surface (X, A) has, in terms of the natural parameterwi, the representation identically equal to one.
As for a Riemann surface,wi will be called the natural parameter near P. Let wibe the natural parameter in a neighborhoodUiof a holomorphic pointP of the quadratic differentialϕonX. Then, using the relation (2.1) it follows that
fwi=
(wi, whenPe∈Ui0 wi, whenPe∈Ui00,
is the natural parameter in the corresponding neighborhood of the pointPe=p(P).
The natural parameter onXC is well defined, because the familyϕeis a quadratic differential onXC, with respect to the analytic atlasAeonXC.
By analogy, fwi is the natural parameter in the corresponding neighborhood of the pointσ(P).e
Proposition 3.2. The quadratic differential ϕe on the Riemann surface (XC,A)e has, in terms offwi, the representation identically equal to one.
Proof. If Pe ∈ Ui0, the differential dfwi becomes dwi = p
ϕ(zi)dzi, therefore by squaring (dwi)2 = ϕ(zi)(dzi)2 so, in terms of fwi = wi, the quadratic differential ϕe has the representation identically equal to one. A similar argument applies if
Pe∈Ui00. The differentialdfwibecomes (dwi)2=ϕ(zi)(dzi)2and in terms offwi=wi, the quadratic differentialϕehas the representation identically equal to one.
Remark 3.3. The quadratic differential ϕe on the Riemann surface XC has, in terms offwi, the representation identically equal to one.
In the special case, whenP is a zero of the quadratic differentialϕ, then as in Strebel [9, Theorem 6.2], we can introduce a local parameter ζi in the neighbor- hood Ui of P, P ↔ ζi = 0, in terms of which the quadratic differential has the representation
(dwi)2=ϕ(zi)(dzi)2= n+ 2 2
2
ζindζi2. (3.1) As a function of ζi, the natural parameter becomes wi = Φ(zi) = (ζi)n+22 and the corresponding natural parameter onXC is
fwi=
((ζi)n+22 , whenPe∈Ui0 (ζi)n+22 , whenPe∈Ui00.
(3.2)
3.1. The ϕ-length of a curve. In this section, we introduce theϕ-metric associ- ated with a quadratic differential on a Klein surface and we study its relation with the correspondingϕ-metric on its canonical (Riemann) double covere XC, see [9].
Letwibe the natural parameter in a neighborhoodUiof a holomorphic pointP of the quadratic differentialϕonX. We define the length element of theϕ-metric,
|dwi|, such that on each connected componentU ofUi∩Uj, the transformation law
|dwi|=
(|dwj|, whenφj◦φ−1i is analytic onφi(U)
|dwj|, whenφj◦φ−1i is antianalytic onφi(U) holds wheneverzi andzj are parameter values near the same pointP ofX. Remark 3.4. The length element of theϕ-metric is a dianalytic invariant, namely
|dwi|=|dwi|=p
|ϕ(zi)||dzi|.
The length element of theϕe-metric is defined by
|dwfi|=
(|dwi|, whenPe∈Ui0
|dwi|, whenPe∈Ui00.
(3.3) LetUibe a neighborhood of a regular pointP of the quadratic differentialϕon X andf−1(Ui) =Ui0 ∪Ui00. Letγbe a piecewise smooth curve inUi. The curveγ has exactly two liftings atf−1(Ui) . Ifγ has the initial pointP andeγis the lifting ofγ atPe∈Ui0, thenσ(eγ) is the lifting of γatPe∈Ui00.
Remark 3.5. Becauseσ: XC → XC is an antianalytic involution, without fixed points, we get that ifeγis the lifting ofγatPe∈Ui00, thenσ(eγ) is the lifting ofγ at Pe∈Ui0.
The curveseγandσ(eγ) are called symmetric onXC. Next, we will identifyeγand σ(eγ), with their images in the complex plane, from the corresponding charts.
The curve γ is mapped by a branch of Φ(zi) =R p
ϕi(zi)dzi onto a curve γ0 in the w-plane. The Euclidean length ofγ0 does not depend on the branch of Φ
which we have chosen. Theϕ-length ofγcan be computed, in terms of the natural parameterwi, by
lϕ(γ) = Z
γ0
|dwi|= Z
γ
p|ϕi(zi)||dzi|, then theϕ-length ofγ is the Euclidean length ofγ0.
Thus, the natural parameter is the local isometry between theϕ- metric and the Euclidean metric.
Remark 3.6. By the definition of the quadratic differential ϕ on X, we obtain that theϕ-length ofγis a dianalytic invariant.
From (3.3), we deduce that theϕ-length ofe γeis l
ϕe(γ) =e
(lϕ(γ), wheneγ∈Ui0 lϕ(γ), wheneγ∈Ui00, wherelϕ(γ) =R
γ
q
|ϕi(zi)||dzi|.
Proposition 3.7. The symmetric curves γe and σ(eγ) have the same ϕ-length,e namely l
ϕe(eγ) = l
ϕe(σ(eγ)) = lϕ(γ). Therefore, the ϕ-metric is a symmetric met-e ric onXC.
Proof. We assume without loss of generality, that eγ ∈Ui0. Then, σ(eγ)∈Ui00. By definition, theϕ-length ofe eγis
l
ϕe(eγ) =lϕ(γ) = Z
γ0
|dwi|= Z
γ
p|ϕi(zi)||dzi| and theϕ-length ofe σ(eγ) is
lϕe(σ(eγ)) =lϕ(γ) = Z
γ0
|dwi|= Z
γ
q
|ϕi(zi)||dzi| whereγ0 is the image ofγby a branch of Φ. Thus,l
ϕe(eγ) =l
ϕe(σ(eγ)).
Next, we consider a point from the surface as being its image through the cor- responding local chart. The ϕ-distance between two pointsz1 and z2 in Ui is, by definition,
dϕ(z1, z2) = inf
γ lϕ(γ) = inf
γ
Z
γ
p|ϕi(zi)||dzi|
where γ ranges over the piecewise smooth curves in Ui joining z1 and z2. The ϕ-distance depends of the domain which is selected.
Because XC is compact, any two points ze1 and ze2 can be connected with a shortest curve whose length is the ϕ-distance betweene ze1 and ze2. We note this distance withd
ϕe(ze1,ze2).
Using the definition of theϕ-length of a curve, we can observe thate dϕe(ze1,ze2) =d
ϕe(ze1,ze2) and d
ϕe(ze1,ze2) =d
ϕe(ze2,ze1).
A piecewise smooth curve is called geodesic if it is locally shortest. The similar notion defined on Riemann surfaces is studied in [9].
A straight arc with respect to the quadratic differentialϕ is a smooth curveγ along which
arg(dw)2= argϕ(z)(dz)2=θ=const., 0≤θ <2π.
Remark 3.8. A straight arc only contains regular points ofϕ.
3.2. The ϕ-metric near holomorphic points. Given a holomorphic quadratic differentialϕ on a Klein surface X, a ϕ-disk is a region which is mapped homeo- morphically onto a disk in the complex plane by a branch of Φ. Theϕ-radius of a ϕ-disk is the Euclidean radius of the corresponding disk in the complex plane.
We want to determine the shortest curve (in the ϕ-metric) near holomorphic points. In this case theϕ-length of a curveγhas sense and is finite, even ifγgoes through a zero of ϕ. Further, we consider a point from the surface X or XC as being its image through the corresponding local charts.
The following theorem is an extension of Strebel’s Theorem 5.4., see [9], from Riemann surfaces to Klein surfaces.
Theorem 3.9. LetP be a regular pointP ofϕ. Then there exists a neighborhoodU ofP such that any two pointsP1andP2inU can be joined by a uniquely determined shortest curveγ(in theϕ-metric) inU. The geodesicγis the pre-image of a straight line segment under a branch ofΦ.
Proof. LetU0 be the largestϕ-disk around a regular pointP. Choose a branch Φ0
of Φ inU0, such that Φ0(P) = 0. If the radius ofU0isr, letV be the disk|w|<r2 andU = Φ−10 (V). Thenw= Φ0(z) =R p
ϕ0(z)dz is the natural parameter inU, whereϕ0(z)(dz)2 is the local representation of the quadratic differential onX.
LetP1 andP2 be two points fromU and wi = Φ0(Pi),i= 1,2. We liftP1 and P2toXC. LetPei andσ(Pei) be the two points ofXC which lie over the same point Pi ofX,i= 1,2.
We notice that γ does not pass through any zeroes of ϕ. If γ is an arbitrary curve joiningP1 andP2 in U, then eithereγ preserves the orientation oreγ changes the orientation in a point of it.
In the first case, eithereγ is contained inU0 oreγ is contained inU00.
If γe is contained in U0, by Strebel’s Theorem 5.4., see [9], fP1 and Pf2 can be joined by a uniquely determined shortest arcγeonXC (in the ϕ-metric). The arce eγ is the pre-image of the straight line segment joining w1 and w2 in thew-plane, under Φ0. Thenlϕ(γ) =l
ϕe(eγ) =R
γ
p|ϕ0(z)||dz|=|w2−w1|.
Similarly, if eγ is contained in U00, both σ(fP1) and σ(fP2) are in U00 and the uniquely determined shortest arcγeis the pre-image of the straight line segment join- ingw1andw2in thew-plane, under Φ0. Thenlϕ(γ) =l
ϕe(σ(γ)) =e R
γ
q
|ϕ0(z)||dz|=
|w2−w1|.
In the second case, becauseγdoes not pass through a zero ofϕ, then we consider the direct analytic continuation of Φ−10 along the straight line segment joining w1
and w2 in thew-plane. Thus, the pointsPf1 and σ(fP2) are joined by the shortest curve eγ with respect to the ϕ-metric one XC, which is composed of straight arcs of the above type and the length of eγ is the sum of the lengths of the component straight arcs. Applying the above results, we obtainlϕ(γ) =l
ϕe(eγ) =|w2−w1|.
In conclusion, if|w2−w1| ≤ |w2−w1|, the geodesic γ is the pre-image of the straight line segment [w1, w2] under Φ0 and if|w2−w1| ≤ |w2−w1|, the geodesic γ is the pre-image of the straight line segment [w1, w2] under a branch of Φ.
Corollary 3.10. The ϕ-distance betweenP1 andP2 , dϕ(z1, z2) = min(|w2−w1|,|w2−w1|)
thusdϕ(z1, z2) = min d
ϕe(ze1,ze2), d
ϕe(ze1,ze2) .
The following theorem is an extension of Strebel’s Theorem 8.1., see [9], from Riemann surfaces to Klein surfaces.
Theorem 3.11. LetP be a zero ofϕof ordern. Then there exists a neighborhood U of P such that any two points P1 and P2 in U can be joined by a uniquely determined shortest curve in U (in the ϕ-metric). The geodesic γ is either the pre-image of a straight line segment under a branch of Φ or is composed of the pre-images of two radii under branches ofΦ.
Proof. LetP be zero ofϕof ordern. Using (3.1), in the neighborhoodU ofP, there is a parameterζ,P ↔ζ= 0 such that the local representation ofϕisϕ(z)(dz)2= (n+22 )2ζn(dζ)2. Thus the corresponding natural parameter is w = Φ(ζ) = ζn+22 . The function Φ maps each one of the sector
{ζ∈C| 2π
n+ 2k≤argζ≤ 2π
n+ 2(k+ 1), k= 0,1, . . . , n+ 1}
onto an upper or lower half-plane.
Theϕ-length of the radius of the circle |ζ|=ρ is equal toρn+22 . Let V be the disk|w|<12ρn+22 and U = Φ−10 (V).
LetP1 andP2 be two points inU andwi= Φ0(Pi),i= 1,2. We liftP1 andP2
toXC. LetPei andσ(Pei) be the two points ofXC which lie over the same pointPi
ofX,i= 1,2.
If γ is an arbitrary curve joining P1 and P2 in U, then either eγ preserves the orientation or eγchanges the orientation in a point of it. We may assume, without loss of generality, thatz16= 0 and argz1= 0.
In the first case, eithereγ is contained inU0 oreγ is contained inU00.
Ifγeis contained in U0, by Strebel’s Theorem 8.1., see [9] and (3.2),Pf1 and fP2
can be joined by a uniquely determined shortest curveeγonXC (in theϕ-metric).e Furthermore,
(a) if |argζe−argζe1| ≤ n+22π for any ζe ∈ eγ, ζe 6= 0, then eγ is the pre-image of the straight line segment joining w1 and w2 in the w-plane, under Φ0. Hence lϕ(γ) =l
ϕe(eγ) =R
γ
p|ϕ0(z)||dz|=|w2−w1|.
(b) If there is a ζe ∈ eγ such that |argζe−argζe1| > n+22π , then the curve eγ is composed, in terms of ζ, of two radii enclosing anglese ≥ n+22π . Hence lϕ(γ) = lϕe(eγ) =|w1|+|w2|.
Analogously, forγecontained inU00, we have:
(c) If |argζe−argζe1| ≤ n+22π for any ζe ∈ eγ, ζe 6= 0, then eγ is the pre-image of the straight line segment joining w1 and w2 in the w-plane, under Φ0. Hence lϕ(γ) =l
ϕe(eγ) =R
γ
q
|ϕ0(z)||dz|=|w2−w1|.
(d) If there is a ζe ∈ eγ such that |argζe−argζe1| > n+22π , then the curve eγ is composed, in terms of ζ, of two radii enclosing angles greater than or equale n+22π . Hencelϕ(γ) =l
ϕe(γ) =e |w1|+|w2|.
In the second case, we have:
(e) If|argζe−argζe1| ≤ n+22π for anyeζ∈γ,e ζe6= 0 we consider the direct analytic continuation of Φ−10 along the straight line segment joining w1 and w2 in the w- plane. The pointsPf1 and σ(Pe2) can be joined by a uniquely determined shortest
curveeγ with respect to theϕ-metric one XC which is composed of straight arcs of the above type. Then,lϕ(γ) =l
ϕe(eγ) =|w2−w1|.
(f) If there is a ζe ∈ eγ such that |argζe−argζe1| > n+22π , then the curve eγ is composed, in terms ofζ, of two radii enclosing angles greater than or equal toe n+22π . Hencelϕ(γ) =l
ϕe(γ) =e |w1|+|w2|.
In conclusion, if|argζe−argζe1| ≤ n+22π , then the geodesicγ is the pre-image of one of the straight line segments, [w1, w2] or [w1, w2], namely the one that has the smallest Euclidean length, under a branch of Φ and if there is a ζe∈ eγ such that
|argζe−argζe1| > n+22π , then the geodesic γ is composed either of the pre-images of the radii [0, w1] and [0, w2] or of the pre-images of the radii [0, w1] and [0, w2], namely those that have the smallest sum of the Euclidean lengths, under branches
of Φ.
Corollary 3.12. The ϕ-distance between the pointsP1 andP2, dϕ(z1, z2) = min |w2−w1|,|w2−w1|,|w1|+|w2|
, thusdϕ(z1, z2) = min(d
ϕe(ze1,ze2), d
ϕe(ze1,ze2))
Acknowledgements. This work was supported by the University of Craiova, [grant number 41C/2014].
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Monica Ros¸iu
Department of Mathematics, University of Craiova, Craiova 200585, Romania E-mail address:monica [email protected]