Dynamics of geodesics for meromorphic connections on Riemann surfaces

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Dynamics of geodesics for

meromorphic connections on

Riemann surfaces

Karim Rakhimov

Supervisor: Prof. Marco Abate

Department of Mathematics

University of Pisa

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Acknowledgements

I would like to show my appreciation to all who supported me while I was working on my Ph.D. research. There are some individuals, however, with whom I interacted more intimately and I’d like to mention them here.

First, I would to thanks to my supervisor Prof. Marco Abate for guiding me in every single step of my research and refining the outcomes throughout the study. The constant and persuasive encouragement of his was essential for the determination of the course of this work. I am closely connected to him for his close and down to earth commitment to the project. His encouragement guided me throughout the whole research and while compiling the thesis. I wouldn’t have anticipated my Ph.D. research to have a better consultant and mentor.

Apart from my supervisor, I won’t forget to express the gratitude to Prof. Azimbai Sadullaev for giving the encouragement and sharing insightful suggestions. Another special thanks goes to Prof. Stefano Luzzatto who recommended me for the position. I want to thank Prof. Sevdiyor Imomkulov for encouraging me to continue with my research in early stages. Their constant guidance throughout my life is impossible to forget.

Several institutions gave me the essential conditions for pursuing such research. Among many, I am especially truly grateful to the University of Pisa’s Department of Mathematics for supporting me. I would like to thank the people for their constant assistance and concern in my work. In par-ticular, I was given so many possibilities by the dynamic systems research group to present my study. I would like to thank Prof. Rita Pardini and Prof. Giovanni Alberti, former and current Ph.D. Directors. I would like to express my sincere appreciation for the courses attended the teachers by both of the Department of Mathematics and the Scuola Normale Superiore. Special recognition goes to the organizers of the seminar on Geometry and Topology at the Weizmann Institute of Science, Israel, which hosted me

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during the development of this thesis. I would like to thank Prof. Sergei Yakovenko in particular for his beneficial discussions and suggestions.

I would also like to thank Dr. Fabrizio Bianchi for his valuable debates and suggestions via Skype and at the 2019 Complex Analysis and Geometry (XXIV) conference in Levico. I really admire his patience and the time he spent responding to my requests.

Last but not least, I would really like to express my gratitude to my family for their presence and their unconditional support.

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Introduction

In this thesis we study the dynamics of geodesics for meromorphic con-nections on Riemann surfaces. Meromorphic concon-nections on Riemann sur-faces have been well studied by many authors (see for example [4, 8, 24]). A meromorphic connection on a Riemann surface S is a C−linear operator ∇ : MT S → M1S⊗ MT S, where MT S is the sheaf of germs of meromorphic

sections of the tangent bundle T S and M1_S is the space of meromorphic 1-forms on S, satisfying the Leibniz rule ∇(f s) = df ⊗s+f ∇s for all s ∈ MT S

and f ∈ MS. A geodesic for a meromorphic connection ∇ is a real smooth

curve σ : I → So_{, where I ⊆ R is an interval and S}o is the complement of the poles of ∇ in S, satisfying the geodesic equation ∇σ0σ0 ≡ 0. To the best

our knowledge, geodesics for meromorphic connections in this sense were first introduced in [8].

0.1 Motivation

One of the main open problems in local dynamics of several complex variables is the understanding of the dynamics in a full neighbourhood of the origin, of holomorphic germs tangent to the identity. In dimension one, the Leau–Fatou flower theorem (see, e.g., [2] or [31]) provides exactly such an understanding. Using this, Camacho ([14]; see also [39]) in 1978 has been able to prove that time-1 maps of homogeneous vector fields provide a complete list of models for the local topological dynamics of one-dimensional holomorphic germs tangent to the identity:

Theorem 0.1.1 (Camacho [14]). Let f (z) = z+aν+1zν+1+. . . with aν+1 6= 0,

be a germ of holomorphic function tangent to the identity. Then f is locally topologically conjugated to the time-1 map of the homogeneous vector field

Q = zν+1 ∂ ∂z

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After 80s of the last century many authors have begun to study the local dynamics of germs tangent to the identity in several complex variables; see, e.g., ´Ecalle [16, 17, 18], Hakim [20, 21], Abate, Bracci, Tovena, Bianchi [1]-[8], Rong [34], Molino [32], Vivas [44], Arizzi, Raissy [9], and others. A few generalizations to several variables of the Leau–Fatou flower theorem have been proved, but none of them was strong enough to be able to describe the dynamics in a full neighbourhood of the origin; furthermore, examples of unexpected phenomena not appearing in the one-dimensional case have been found. Thus it is only natural to try and study the dynamics of mean-ingful classes of examples, with the aim of extracting ideas applicable to a general setting; and Camacho’s theorem suggests that a particularly inter-esting class of examples is provided by time-1 maps of homogeneous vector fields. Actually, the evidence collected so far strongly suggests that a several variable version of Camacho’s theorem might hold: generic germs tangent to the identity should be locally topologically conjugated to time-1 maps of homogeneous vector fields. This is the approach initiated in [8], where the authors discovered that there is a strong relationship between the dy-namics of the time-1 map of homogeneous vector fields and the dydy-namics of geodesics for meromorphic connections on Riemann surfaces. To describe this relationship, we need to introduce a few notations and definitions.

Let Q be a homogeneous vector field on Cn of degree ν + 1 ≥ 2. First of all, notice that the orbits of its time-1 map are contained in the real integral curves of Q; so we are interested in studying the dynamics of the real integral curves of the complex homogeneous vector field Q (actually, it turns out that complex integral curves of a homogeneous vector field are related to—classically studied—sections which are horizontal with respect to a meromorphic connection; see [8] for details).

A characteristic direction for Q is a direction v ∈ Pn−1_{(C) such that the}

complex line (the characteristic leaf) issuing from the origin in the direction v is Q−invariant. An integral curve issuing from a point of a characteristic leaf stays in that leaf forever; so the dynamics in a characteristic leaf is one-dimensional, and thus completely known. In particular, if the vector field Q is a multiple of the radial field (we shall say that Q is dicritical ) every direction is characteristic; thus the dynamics is one-dimensional and completely understood. So, we are mainly interested in understanding the dynamics of integral curves outside the characteristic leaves of non-dicritical vector fields. In [8] Abate and Tovena proved the following result:

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Theorem 0.1.2 (Abate and Tovena [8]). Let Q be a non-dicritical homoge-neous vector field of degree ν + 1 ≥ 2 in Cn and let MQ be the complement

in Cn of the characteristic leaves of Q. Let [·] : Cn\ {O} → Pn−1

(C) denote the canonical projection. Then there exists a singular holomorphic foliation F of Pn−1_{(C) in Riemann surfaces, and a partial meromorphic connection ∇}

inducing a meromorphic connection on each leaf of F , whose poles coincide with the characteristic directions of Q, such that the following holds:

• if γ : I → MQ is an integral curve of Q then the image of [γ] is

contained in a leaf S of F and it is a geodesic for ∇ in S; • conversely, if σ : I → Pn−1

(C) is a geodesic for ∇ in a leaf S of F then there are exactly ν integral curves γ1, ..., γν : I → MQ such that

[γj] = σ for j = 1, ..., ν.

Due to this result, we see that the study of integral curves for a homoge-neous vector field in Cn_{is reduced to the study of geodesics for meromorphic}

connections on a Riemann surface S.

0.2 Poincar´

e-Bendixson theorems

In [8], Abate and Tovena studied the ω-limit sets of the geodesics of meromorphic connections on P1(C). They gave complete classification of the ω-limit sets of simple geodesics. Later, in [4], Abate and Bianchi studied the same problem for any compact Riemann surface S and they proved Poincar´ e-Bendixson theorems for simple geodesics. Since everything has done for simple geodesic the only question left is “what happens if a geodesic intersects itself infinitely many times?”. For the case P1_{(C) Abate conjectured the}

following

Conjecture 0.2.1 (Abate). Let σ : [0, ε) → So _{be a maximal geodesic for a}

meromorphic connection ∇ on P1(C), where So = P1(C)\{p0, p1, . . . , pr} and

p0, p1, . . . , pr are the poles of ∇. If σ intersects itself infinitely many times

then the ω-limit set of σ is P1_(C).

0.3 Main results

Before stating the main results of the thesis we need to introduce a few notations and definitions. Let ∇ be a meromorphic connection on a Riemann

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surface S. Let {(Uα, zα)} be an atlas for S. It is not difficult to see that

there exists ηα ∈ M1S(Uα) on Uα such that ∇

∂ ∂zα = ηα ⊗ _∂z∂_α, where ∂

∂zα is the induced local generator of T S over Uα. We shall say ηα is the

local representation of ∇ on Uα. A pole p is said to be Fuchsian if the local

representation of ∇ on one (and hence any) chart (U, z) around p has a simple pole at p. If all poles of ∇ are Fuchsian then we shall say ∇ is Fuchsian. Let So _{be the complement of the poles of ∇ and G a multiplicative subgroup}

of C∗. We say ∇ has monodromy in G if there exists an atlas {(Uα, zα)}

for So _{such that the representations of ∇ on U}

α’s are identically zero and

the transition functions are of the form zβ = aαβzα + cαβ on Uα ∩ Uβ, with

aαβ ∈ G and cαβ ∈ C. We say ∇ has real periods if G ⊂ S1.

To state the first main result we need to introduce the notion of meromor-phic k-differentials. A meromormeromor-phic k-differential q on a Riemann surface S is a meromorphic section of the k-th power of the canonical line bundle. The zeros and the poles of q constitute the set Σ of critical points of q. It is not difficult to see that a k-differential q given locally as q = q(z)dz2 _(for

more details see Definition 3.3.1). Then there is a metric g locally given as g12 = |q(z)|

1

k|dz|. In particular, g is a flat metric on So:= S \ Σ (see

Propo-sition 3.3.2). Finally, a smooth curve σ : [0, ε) → So is a geodesic for q if it is a geodesic for g.

When k = 1 we get the meromorphic Abelian differential which are deeply studied both from a geometrical and a dynamical point of view. Theory of translation surfaces (corresponding to Abelian differentials) provides new insights in the study of the dynamics of billiards through the methods of al-gebraic geometry and renormalization theory. When k = 2 we get the mero-morphic quadratic differential which are a well studied subject in Teichm¨uller theory. Extracting the k-th root, one can think of an arbitrary k-differential as a multi-valued meromorphic Abelian differential on S. In general, differ-entials of order k > 2 are much less studied than their quadratic counter-part. Nevertheless a complete classification of the ω-limit sets of geodesics for meromorphic k-differentials on compact Riemann surfaces is already given (see [41]).

A way of studying meromorphic connections is to find a relation with meromorphic k-differentials. We say a meromorphic k-differential q and meromorphic connection ∇ are adapted to each other if they have the same geodesics and the same critical points. Our first main theorem describes an exact relation between meromorphic k-differentials and meromorphic

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con-nections.

Theorem 0.3.1. Let ∇ be a Fuchsian meromorphic connection on a Rie-mann surface S. If ∇ has monodromy in Zk and residues in 1_kZ then there

is a meromorphic k-differential q adapted to ∇ (here we identify Zk with the

multiplicative group of k-th roots of unity). Moreover, q is unique up to a non-zero constant multiple.

On the other hand, if q is a meromorphic k-differential on a Riemann surface S then there exists a unique meromorphic connection ∇ adapted to q. Moreover, ∇ is Fuchsian and it has monodromy in Zk and residues in 1_kZ.

Thanks to Theorem 0.3.1 we get a complete classification of the ω-limit sets of geodesics for Fuchsian meromorphic connections with monodromy in Zk on compact Riemann surfaces. In particular, we have a classification for

the ω-limit sets of infinite self-intersecting geodesics. Another consequence of Theorem 0.3.1 is the proof of Conjecture 0.2.1 for Fuchsian meromorphic connections with monodromy in Zk.

Theorem 0.3.2. Let ∇ be a Fuchsian meromorphic connection on P1_(C)

with monodromy in Zk. Then Conjecture 0.2.1 is true.

In [8] and [4] it is only given the possible classifications of the ω-limit sets of simple geodesics. But there were not given anything about infinite self-intersecting geodesics. The next main result describes the possible classifica-tions of the ω-limit sets of infinite self-intersecting geodesics of meromorphic connections having real part of residues in 1_k_Z.

Theorem 0.3.3. Let ∇ be a meromorphic connection on a compact Riemann surface S with residues in 1_k_{Z + iR for some k ∈ N}∗. Set So = S \ Σ, where Σ is the set of poles of ∇. Let σ : [0, ε) → So _{be a maximal geodesic of ∇.}

Assume σ intersects itself infinitely many times. Then either

1. the ω-limit set of σ in S is given by the support of a (possibly non-simple) closed geodesic; or

2. the ω-limit set of σ in S is a graph of (possibly self-intersecting) saddle connections; or

3. the ω-limit set of σ has non-empty interior and non-empty bound-ary, and each component of its boundary is a graph of (possibly self-intersecting) saddle connections with no spikes and at least one pole; or

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4. the ω-limit set of σ in S is all of S.

Actually, we do not have any examples for the cases 1-3. Hence to prove Conjecture 0.2.1 for this case we just have to show that the cases 1-3 do not appear.

In [8] and [4] it is shown that two possibilities for the ω-limit set of a simple geodesic are a closed geodesic and a graph of saddle connections. The next main result of the thesis shows that it is impossible for Fuchsian meromorphic connections with real periods.

Theorem 0.3.4. Let ∇ be a Fuchsian meromorphic connection with real periods on a compact Riemann surface S. Set So := S \ Σ, where Σ is the set of poles of ∇. Let σ : [0, ε) → So _{be a maximal simple non-periodic geodesic}

for ∇. Then the ω-limit set of σ is neither a periodic geodesic nor a graph of saddle connections.

To study the ω-limit set of a geodesic of a meromorphic connection ∇ on a Riemann surface S it is useful to know the local behavior of geodesics around a pole p. The next proposition gives such an understanding around Fuchsian poles with real part of residue less than or equal to −1.

Proposition 0.3.5. Let ∇ be a meromorphic connection on a Riemann sur-face S. Set So _{:= S \ Σ where Σ is the set of poles for ∇. Let σ : [0, ε) → S}o

be a maximal geodesic of ∇ and W its ω-limit set. Let p be a Fuchsian pole with Re Resp∇ ≤ −1. If p ∈ W then W = {p}.

A curve σ : [0, ε) → S on a compact Riemann surface S is said to be minimal if it is dense on S. Hence Proposition 0.3.5 shows that to study minimality of the geodesics of Fuchsian meromorphic connections on compact Riemann surfaces it is only natural to study meromorphic connections with real part of residues are greater than −1. In the next main result of the thesis we studied minimality of the geodesics of Fuchsian meromorphic connections with residues are real and greater than −1.

Theorem 0.3.6. Let ∇ be a Fuchsian meromorphic connection on P1_(C)

and Σ = {p0 = ∞, p1, ..., pr} be the set of poles of ∇. Set So = P1(C) \ Σ.

Assume the residues of ∇ are real and greater than −1. Pick a point p ∈ So_.

Then for almost all θ ∈ S1 _{a maximal geodesic starting at p with θ-direction}

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Of course, Theorem 0.3.6 does not directly say that Conjecture 0.2.1 is almost surely true for Fuchsian meromorphic connections with real residues. But the next two propositions show that it is actually true.

Proposition 0.3.7. Let ∇ be a meromorphic connection with residues in

1

2Z + iR on a compact Riemann surface S. Then there is no self intersecting

geodesics for ∇.

Since the ω-limit sets of simple geodesics completely classified in [8] and [4] Proposition 0.3.7 implies that everything has done for meromorphic con-nections with residues in 1₂_{Z + iR.} Furthermore, a possibility of infinite self-intersection of geodesics is studied in the next proposition.

Proposition 0.3.8. Let ∇ be a meromorphic connection on a compact Rie-mann surface S. Set So _{:= S \ Σ, where Σ is the set of poles of ∇. Let p be}

a Fuchsian critical point for ∇ with residue in R \ 12Z. Let σ : [0, ε) → S o

be a maximal geodesic and W its ω-limit set. If p is an interior point of W then σ intersect itself infinitely many times.

As a consequence we can see that if there is a pole with residue in R \ 12Z

then any minimal geodesic in P1(C) intersects itself infinitely many times. Hence Theorem 0.3.6 implies that Conjecture 0.2.1 is almost surely true for Fuchsian meromorphic connections with real residues.

0.4 Overview of the thesis

In the first chapter we recall some definitions and facts on meromorphic connections on Riemann surfaces.

In Section 2.1, we introduce the notion of singular flat metrics and we present general properties of singular flat metrics. In Section 2.2, we study relation between singular flat metrics and meromorphic connections.

In Section 3.1, we recall some definitions and fundamental results on the theory of meromorphic quadratic differentials and we review a relation be-tween meromorphic quadratic differentials and singular flat metrics. In Sec-tion 3.2, we recall the noSec-tion of argument for a quadratic differential along a smooth curve σ. Moreover, Section 3.2 includes some fundamental results like uniquely ergodicity of a quadratic differentials on compact Riemann sur-face, and relation between interval exchange transformations and quadratic differentials. Section 3.3 covers material about meromorphic k-differentials.

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In Section 3.4, we describe a relation between meromorphic connections and meromorphic k-differentials and we prove Theorem 0.3.1. As a consequence we prove Proposition 0.3.5 and a conjecture proposed by Abate and Tovena (see Conjecture 3.4.5).

In Section 4.1, we introduce the notion of ∇-chart and ∇-atlas and we study some properties of ∇-charts. In Section 4.2, we introduce the notion of meromorphic differential and we study relation between meromorphic G-differentials and meromorphic connections. Furthermore, we introduce the notion of argument along geodesics of a meromorphic G-differentials. By using it we find a subset of the set of meromorphic connections such that all geodesics of the meromorphic connections are simple. In particular, we prove Proposition 0.3.7. In Section 4.3, we introduce the notion of canon-ical covering for a meromorphic connection on a compact Riemann surface S and we prove Theorem 0.3.3. In Section 4.4, we study sequences of mero-morphic connections. We also introduce the notion of uniform convergence of a sequence of meromorphic connections and we study some properties of uniformly convergent sequences of meromorphic connections.

In Chapter 5, we study local behaviour around non-resonant Fuchsian poles. In Section 5.1, we show the existence of a distinguished chart around a non-resonant Fuchsian pole of a meromorphic connection ∇. In Section 5.2, we study local behaviour of geodesics of a meromorphic connection ∇ around non-resonant Fuchsian poles. Furthermore, we prove Proposition 0.3.8.

In Chapter 6, we prove analogs of some important equalities proved in [8] and [4] (see (1.7) and (1.9)). By using the equalities we study uniqueness of geodesic connecting two points for a Fuchsian meromorphic connection with real residues greater than −1. In Section 6.1, we continue the local study of singular flat metrics adapted to a meromorphic connection in a neighbourhood of a Fuchsian pole with residue greater than −1. In Section 6.3, we study uniqueness of simple geodesics of a meromorphic connection ∇ connecting two points of a compact Riemann surface S.

In Section 7.1, we study relation between monodromy group and residues of a meromorphic connection ∇ on a compact Riemann surface S. In Section 7.2, we study simple periodic geodesics and graphs of saddle connections. In particular, we show that the ω-limit set of a non-periodic simple geodesic of a meromorphic connection with real periods is neither a periodic geodesic nor a graph of saddle connections and we prove Theorem 0.3.4. Finally, in Section 7.3, we prove a version of the Poincar´e-Bendixson theorem and Theorem 0.3.2.

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In Section 8.1, we study minimality of the geodesics of a Fuchsian mero-morphic connection ∇ with real periods and residues greater than −1. In Section 8.2, we introduce the notion of Zk-approximability by using it we

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Chapter 1 Meromorphic connections

This chapter introduces basic notation and recalls results that will be used throughout this work. In Section 1.1, we review some of the standard facts on the theory of holomorphic connections over a complex line bundle L on a Riemann surface S. In Section 1.2, we repeat definition of a meromorphic connection on a Riemann surface S and we study some of its properties. Finally, the chapter ends with Section 1.3 that recalls some fundamental results of the theory of dynamics of geodesics for meromorphic connections on compact Riemann surfaces and local behaviors of geodesics around Fuchsian poles.

1.1 Holomorphic connections

In this section we repeat some definitions and theorems from [8] and [4]. Definition 1.1.1. Let L be a complex line bundle on a Riemann surface S. A holomorphic connection on L is a C−linear map ∇ : L → Ω1S⊗L satisfying

the Leibniz rule

∇(se) = ds ⊗ e + s∇e

for all s ∈ OS and e ∈ L, where L denotes the sheaf of germs of holomorphic

sections of L, while OS is the sheaf of germs holomorphic functions on S and

Ω1

S is the sheaf of germs of holomorphic 1-forms on S.

Let us see what this condition means in local coordinates. Let ∇ be a holomorphic connection on L. Let {(Uα, zα, eα)} be an atlas of S trivializing

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of L|Uα. It is not difficult to see that there exists a holomorphic 1−form

ηα ∈ Ω1S(Uα) on Uα such that

∇eα = ηα⊗ eα. (1.1)

Definition 1.1.2. We say ηα the local representation of ∇ on Uα.

Now, for given local representations {ηα} we look for a condition which

guarantees the existence of a holomorphic connection ∇. Let {ξαβ} be the

cocycle representing the cohomology class ξ ∈ H1_{(S, O}∗_{) of L; over U} α∩ Uβ we have eβ = eαξαβ and thus ∇(eβ) = ∇(eαξαβ) ⇔ ηβ⊗ eβ = ξαβηα⊗ eα+ dξαβ⊗ eα if and only if ηβ = ηα+ dξαβ ξαβ (1.2) Recalling the short exact sequence of sheaves

0 → C∗ → O∗ ∂ log−−→ Ω1_S → 0

we can see that equality (1.2) shows that the existence of a holomorphic connection ∇ is equivalent to the vanishing of the image of ξ under the map ∂ log : H1_{(S, O}∗_{) → H}1_{(S, Ω}1

S) induced on cohomology. Hence the class ξ is

the image of a class ˆξ ∈ H1_{(S, C}∗_{). We recall how to find a representative ˆ}_ξ αβ

of ˆξ. After shrinking the Uα’s, if necessary, we can find holomorphic functions

Kα ∈ O(Uα) such that ηα = ∂Kα on Uα. Set

ˆ ξαβ =

exp(Kα)

exp(Kβ)

ξαβ. (1.3)

in Uα ∩ Uβ. Then (1.2) implies that ˆξαβ is a complex non-zero constant

defining a cocycle representing ξ.

Definition 1.1.3. The homeomorphism ρ : π1(S) → C∗corresponding to the

class ˆξ under the canonical isomorphism H1_{(S, C}∗_{) ∼}

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Hom(π1(S), C∗) is the monodromy representation of the holomorphic

con-nection ∇. We shall say that ∇ has monodromy in G, a multiplicative

subgroup of C∗, if the image of ρ is contained in G, that is if ˆξ is the image of a class in H1_{(S, G) under the natural inclusion G ,→ C}∗_{. We say G is}

a monodromy group of ∇. We shall say that ∇ has real periods if it has monodromy in S1.

Remark 1.1.4. In [8] it is introduced a period map ρ0 : H1(S, Z) → C

associ-ated to ∇ and it has the following relation ρ = exp(2πiρ0)

with the monodromy representation ρ. Therefore ∇ has real periods if and only if the image of the period map is contained in R.

It is well known that to a Hermitian metric g on a complex vector bundle over a complex manifold M can be associated a connection ∇ (not necessarily holomorphic) such that ∇g ≡ 0, the Chern connection of g. The converse problem was also studied by Abate and Tovena for holomorphic connections: given a holomorphic connection ∇, does there exist a Hermitian metric g so that ∇g ≡ 0?

Definition 1.1.5. Let ∇ be a holomorphic connection on a line bundle L over a Riemann surface S. We say that a Hermitian metric g on L is adapted to ∇ if ∇g ≡ 0, that is if

X(g(R, T )) = g(∇XR, T ) + g(R, ∇XT )

and

X(g(R, T )) = g(∇_XR, T ) + g(R, ∇XT )

for all sections R, T of L, and all vector fields X on S.

As usual, let us check the condition in local coordinates. Let {(Uα, zα, eα)}

be an atlas trivializing L. A Hermitian metric g on L is locally represented by a positive C∞ function nα ∈ C∞(Uα, R+) given by

nα = g(eα, eα).

Then we can see that ∇g ≡ 0 over Uα if and only if

∂nα = nαηα, (1.4)

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Proposition 1.1.6 ([8, Proposition 1.1]). Let L be a complex line bundle on a Riemann surface S, and ∇ a holomorphic connection on L. Let (Uα, zα, eα)

be a local chart trivializing L, and define ηα ∈ Ω1S(Uα) by setting ∇eα =

ηα ⊗ eα. Assume that we have a holomorphic primitive Kα of ηα on Uα.

Then

nα = exp(2Re Kα) (1.5)

is a positive solution of (1.4). Conversely, if nα is a positive solution of

(1.4) then for any z0 ∈ Uα and any simply connected neighborhood U ⊆ Uα

of z0 there is a holomorphic primitive Kα ∈ O(U ) of ηα over U such that

nα = exp(2ReKα) in U . Furthermore, Kα is unique up to a purely imaginary

additive constant. Finally, two (local) solutions of (1.4) differ (locally) by a positive multiplicative constant.

It is not difficult to see that Gaussian curvature of the local metrics (1.5) is identically zero. The proposition shows that for any holomorphic connection ∇ we can always associate local metrics g adapted to ∇. A global metric adapted to ∇ might not exist (see [8, Proposition 1.2]). Existence of a global metric adapted to a holomorphic connection ∇ on a line bundle L is completely studied in [8].

Theorem 1.1.7 ( [8, Proposition 1.2]). Let L be a complex line bundle on a Riemann surface S and ∇ a holomorphic connection on L. Then there exists a Hermitian metric adapted to ∇ if and only if ∇ has real periods.

1.1.1 Holomorphic connections on the tangent bundle

Let S be a Riemann surface and ∇ a holomorphic connection on the tangent bundle T S. For simplicity we just say ∇ is a holomorphic connection on S instead of T S.

Definition 1.1.8. A geodesic for a holomorphic connection ∇ on S is a real curve σ : I → S, with I ⊆ R an interval, such that ∇σ0σ0 ≡ 0, where

∇us := ∇s(u).

Let {(Uα, zα)} be an atlas for S and σ : I → Uα, with I ⊆ R an interval,

a smooth curve. Then σ is a geodesic for a meromorphic connection ∇ if and only if

σ00(t) + (fα◦ σ)σ02 ≡ 0

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1.1.2 Local isometries

In this subsection we introduce the notion of local isometry of a holo-morphic connection on a simply connected chart. Moreover, we study some properties of local isometries and we often use this notion in the rest of the thesis.

Definition 1.1.9. Let ∇ be a holomorphic connection on a Riemann surface S. Let (Uα, zα) be a simply connected chart for S. Let ηα be the

representa-tion of ∇ on Uα, and Kα : Uα → C a holomorphic primitive of ηα. Finally,

let Jα : Uα→ C be a holomorphic primitive of exp(Kα). We say Jα is a local

isometry of ∇ on Uα.

Theorem 1.1.10 ([8, Proposition 2.2]). Let ∇ be a holomorphic connection on a Riemann surface S. Let (Uα, zα) be a simply connected chart for S.

Let Jα : Uα → C be a local isometry of ∇ on Uα. Then a smooth curve

σ : I → Uα is a geodesic for ∇ if and only if there are c0, w0 ∈ C such

that Jα(σ(t)) = c0t + w0. In particular, the geodesic with σ(0) = z0 and

σ0(0) = v0 ∈ C∗ is given by

σ(t) = J_α−1(c0t + Jα(z0)),

where c0 = exp(Kα(z0))v0 and Jα−1 is the analytic continuation of the local

inverse of Jα near Jα(z0) such that Jα−1(Jα(z0)) = z0.

Finally, a curve σ : [0, ε) → Uα is a geodesic if and only if

σ0(t) = exp(−Kα(σ(t))) exp(Kα(σ(0)))σ0(0),

if and only if

Jα(σ(t)) = exp(Kα(σ(0)))σ0(0)t + Jα(σ(0)).

Lemma 1.1.11. Let ∇ be a holomorphic connection on a Riemann surface S. Let (U, z) be a simply connected chart for S. Let J1 and J2 be local

isometries of ∇ on U . Then J1 and J2 are linear dependent, i.e., there exist

a ∈ C∗ and b ∈ C such that J1 ≡ aJ2+ b.

Proof. Let η be the representation of ∇ on U . Let K1 and K2be holomorphic

primitives of η on U . It is easy to see that there exists c ∈ C such that K1 ≡ K2 + c. Set Fj = eKj for j = 1, 2. Then F1 = ecF2. Let Jj be a

holomorphic primitive of Fj. It is not difficult to see that there exists b ∈ C

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Let S be a Riemann surface, ∇ a holomorphic connection on S. Let π : ˜S → S be the universal covering map. Let ˜∇ = π∗_{∇ be the holomorphic}

connection on ˜S induced by ∇ via π. The following theorem shows the relation between the geodesics of ∇ and the geodesics of ˜∇.

Theorem 1.1.12 ([8, Proposition 3.1]). Let S be a Riemann surface, and ∇ a holomorphic connection on S. Let π : ˜S → S be the universal covering map, and ˜∇ the holomorphic connection on ˜S induced by ˜∇ via π. Then a curve ˜σ : I → S is a geodesic for ˜∇ if and only if σ = π ◦ ˜σ is a geodesic for ∇.

In [8], it was computed the monodromy group when S ⊆ C, that is when S is covered by a single chart. The most interesting case will be when S is the complement in P1_{(C) of a finite set of points.}

Proposition 1.1.13 ([8, Proposition 3.6]). Let S ⊆ C be a (multiply con-nected) domain, ∇ a holomorphic connection on T S, and η the holomorphic 1-form representing ∇. Then the monodromy representation ρ : H1(S, Z) →

C∗ is given by ρ(γ) = exp Z γ η for all γ ∈ H1(S, Z).

1.2 Meromorphic connection

We will now define meromorphic connection

Definition 1.2.1. A meromorphic connection on the tangent bundle T S of a Riemann surface S is a C-linear map ∇ : MT S → M1S⊗ MT S satisfying

the Leibniz rule

∇( ˜f ˜s) = d ˜f ⊗ ˜s + ˜f ∇˜s

for all ˜s ∈ MT S and ˜f ∈ MS, where MT S denotes the sheaf of germs

meromorphic sections of T S, while MS is the sheaf of germs of meromorphic

functions and M1_S is the sheaf of meromorphic 1-forms on S.

Let (Uα, zα) be a local chart for S, and ∇ a meromorphic connection on

S. Then there exists ηα ∈ M1S(Uα), such that

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where ∂α := _∂z∂_α is the induced local generator of T S over Uα. Similarly as

in holomorphic connection we have ηβ = ηα+

1 ξαβ

∂ξαβ (1.6)

on Uα ∩ Uβ, where ξαβ := ∂z_∂zα

β. In particular, if all the representations are

holomorphic then ∇ is a holomorphic connection. We say p ∈ S is a pole for a meromorphic connection ∇ if p is a pole of ηα for some (and hence any)

local chart Uα at p. If Σ is the set of poles of ∇, then ∇ is a holomorphic

connection on So = S \ Σ. So we can define notion of geodesic for ∇ on So as in holomorphic connection.

Definition 1.2.2. A geodesic for a meromorphic connection ∇ on T S is a real curve σ : I → So_{, with I ⊆ R an interval, such that ∇}σ0σ0 ≡ 0.

Definition 1.2.3. The residue Resp∇ of a meromorphic connection ∇ at

a point p ∈ S is the residue of any 1-form ηα representing ∇ on a local

chart (Uα, zα) at p. The set of all residues of ∇ is denoted by Res∇, i.e.,

Res∇ := {Resp∇ : p ∈ S} \ {0}.

By condition (1.6) the residue of ∇ does not depend on the choice of charts. Definition 1.2.4. We say that p ∈ S is a Fuchsian pole of a meromorphic connection ∇ if there exists a (and hence any) chart (Uα, zα) around p such

that the representation of ∇ has a simple pole at p. If all poles of ∇ are Fuchsian then we say ∇ is a Fuchsian meromorphic connection.

1.3 Meromorphic connections on compact

Riemann surfaces

In this section we recall some important results from [8] and [4]. Let first ∇ be a meromorphic connection on P1_{(C); we can consider it as a}

holomorphic connection on S = P1_{(C) \ {p}

0, ..., pr}, where {p0, ..., pr} are the

poles of the meromorphic connection. Without loss of generality, we shall always assume p0 = ∞, so that S ⊆ C.

Definition 1.3.1. Let ∇ be a meromorphic connection on P1_{(C), with poles}

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is a simply connected domain R0 ⊂ P1(C) whose boundary is composed by

s ≥ 1 simple smooth curves σj : [0, lj] → P1(C) such that σj |(0,lj) is a

geodesic with σj(lj) = σj+1(0) = zj+1 for j = 1, ..., s − 1, σs(ls) = σ1(0) = z1

and no other intersections are allowed; the curves are listed so that ∂R0 is

positively oriented (that is R0 is the interior of ∂R0). The points z1, ..., zs

are the vertices of R0. We say R0 is a regular geodesic polygon if all vertices

of R0 are regular points. We say R0 is a Fuchsian geodesic polygon if any

vertex of R0 is either a Fuchsian pole or a regular point for ∇.

By applying Gauss-Bonnet theorem to the local metrics adapted to ∇ the next important equality is proven in [8].

Theorem 1.3.2 ([8, Theorem 4.1]). Let ∇ be a meromorphic connection on P1(C), with poles {p0 = ∞, p1, ..., pr}, and set S = P1(C) \ {p0, ..., pr} ⊆ C.

Let R0 ⊂ P1(C) be an s−sided regular geodesic polygon with vertices z1, ..., zs.

For j = 1, ..., s let εj ∈ (−π, π) be the external angle in zj, and let {p1, ..., pg}

be the poles of ∇ contained in R0. Then

s X j=1 εj = 2π 1 + g X j=1 Re Respj(∇) ! . (1.7)

Corollary 1.3.3 ([8, Corollary 4.3]). Let ∇ be a meromorphic connection on P1_{(C), with poles {p}

0 = ∞, p1, ..., pr}, and set S = P1(C) \ {p0, ..., pr} ⊆

C. Let σ0 : [0, l0] → S and σ1 : [0, l1] → S be two distinct geodesics with

σ0(0) = z0 = σ1(0) and σ0(l0) = z1 = σ1(l1) and not intersecting elsewhere.

Let {p1, ..., pg} be the poles of ∇ contained in the simply connected domain R0

bounded by σ0 and σ1, and εj ∈ (−π, π) the external angle at zj, for j = 1, 2.

Then ε1+ ε2 = 2π 1 + g X j=1 Re Respj(∇) ! . (1.8)

Let now S be any compact Riemann surface. Then the analogs of the results above are proven in [4].

Definition 1.3.4. Let S be a compact Riemann surface. Let ∇ be a mero-morphic connection on S and let So _{⊆ S be the complement of the poles.}

• A geodesic (n-)cycle is the union of n simple smooth curves σj : [0, 1] →

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σj(0) = σj−1(1) for j = 2, ..., n and σ1(0) = σn(1). The points σj(0)

will be called vertices of the geodesic cycle. We say a geodesic cycle is regular if all vertices of the geodesic cycle are regular points. We say a geodesic cycle is Fuchsian if any vertex the geodesic cycle is either a Fuchsian pole or a regular point.

• A (m-)multicurve is a union of m disjoint geodesic cycles. A multicurve will be said to be disconnecting if it disconnects S, non-disconnecting otherwise. We say a m-multicurve is Fuchsian (regular ) if it is a union of m disjoint Fuchsian (regular) geodesic cycles.

• A part P is the closure of a connected open subset of S whose boundary is a multicurve γ. A component σ of γ is surrounded if the interior of P contains both sides of a tubular neighbourhood of σ in S; it is free otherwise. The filling ˆP of a part P is the compact surface obtained by gluing a disk along each of the free components of γ, and not removing any of the surrounded components of γ .

Theorem 1.3.5 ([4, Theorem 3.1]). Let ∇ be a meromorphic connection on a compact Riemann surface S, with poles {p1, ..., pr} and set So = S \

{p1, ..., pr}. Let P be a part of S whose boundary multicurve γ ⊂ So is regular

an it has mf ≥ 1 free components, positively oriented with respect to P . Let

z1, ..., zs denote the vertices of the free components of γ, and εj ∈ (−π, π)

the external angle at zj. Suppose that P contains the poles {p1, ..., pg} and

denote by g_Pˆ the genus of the filling ˆP of P . Then

s X j=1 εj = 2π 2 − mf − 2g_Pˆ + g X j=1 Re Respj(∇) ! . (1.9)

Corollary 1.3.6 ([4, Corollary 3.2]). Let ∇ be a meromorphic connection on a compact Riemann surface S, with poles {p1, ..., pr} and set So = S \

{p1, ..., pr}. Let γ be a disconnecting regular geodesic 2-cycle. Let P be one

of the two parts in which S is disconnected by γ, and ε0, ε1 ∈ (−π, π) the two

external angles of γ. Then

ε1+ ε2 = 2π  1 − 2g_Pˆ + X pj∈P Re Respj(∇)  . (1.10)

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1.3.1 Poincar´

e-Bendixson theorems

In this subsection we recall Poincar´e-Bendixson theorems for meromor-phic connections on compact Riemann surfaces, i.e., a classification of the possible ω-limit sets for the geodesics of meromorphic connections on com-pact Riemann surfaces.

Definition 1.3.7. Let σ : (ε−, ε+) → S be a curve in a Riemann surface S.

Then the ω-limit set of σ is given by the points p ∈ S such that there exists a sequence {tn}, with tn % ε+, such that σ(tn) → p. Similarly, the α-limit

set of σ is given by the points p ∈ S such that there exists a sequence {tn},

with tn& ε−, such that σ(tn) → p.

Definition 1.3.8. A geodesic σ : [0, l] → S is closed if σ(l) = σ(0) and σ0(l) is a positive multiple of σ0(0) ; it is periodic if σ(l) = σ(0) and σ0(l) = σ0(0).

Definition 1.3.9. Let σ : I → S be a curve in S = P1_{(C) \ {p}

0, ..., pr}. A

simple loop in σ is the restriction of σ to a closed interval [t0, t1] ⊆ I such

that σ|[t0,t1] is a simple loop γ. If p0, ..., pg are the poles of ∇ contained in

the interior of γ, we shall say that γ surrounds p0, ..., pg.

Definition 1.3.10. A saddle connection for a meromorphic connection ∇ on S is a maximal geodesic σ : (ε−, ε+) → S \ {p0, ..., pr} (with ε− ∈ [−∞, 0)

and ε+ ∈ (0, +∞]) such that σ(t) tends to a pole of ∇ both when t ↑ ε+ and

when t ↓ ε−.

A graph of saddle connections is a connected graph in S whose vertices are poles and whose arcs are disjoint saddle connections. A spike is a saddle connection of a graph which does not belong to any cycle of the graph.

A boundary graph of saddle connections is a graph of saddle connections which is also the boundary of a connected open subset of S. A boundary graph is disconnecting if its complement in S is not connected.

Next we state the Poincar´e-Bendixson theorem for the meromorphic con-nections on any compact Riemann surface S which proved in [8, Theorem 4.6] and [4, Theorem 4.3].

Theorem 1.3.11 (Abate-Bianchi-Tovena). Let σ : [0, ε) → So _{be a maximal}

geodesic for a meromorphic connection ∇ on S, where So = S\{p0, p1, . . . , pr}

and p0, p1, . . . , pr are the poles of ∇. Then either

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2. σ is closed; or

3. the ω-limit set of σ in S is given by the support of a closed geodesic; or 4. the ω-limit set of σ in S is a boundary graph of saddle connections; or 5. the ω-limit set of σ in S is all of S; or

6. the ω-limit set of σ has non-empty interior and non-empty boundary, and each component of its boundary is a graph of saddle connections with no spikes and at least one pole; or

7. σ intersects itself infinitely many times.

Furthermore, in cases 2 or 3 the support of σ is contained in only one of the components of the complement of the ω-limit set, which is a part P of S having the ω-limit set as boundary.

In particular, if S = P1_{(C) then}

1. if σ is closed then σ surrounds poles p1, ..., pg with

g

X

j=1

Re Respj(∇) = −1; (1.11)

2. if the ω-limit set of σ in P1_{(C) is given by the support of a closed}

geodesic then the closed geodesic surrounds poles p1, ..., pg with (1.11);

3. if the ω-limit set of σ in P1(C) is a cycle of saddle connections then the cycle of saddle connections surrounds poles p1, ..., pg with (1.11);

4. if σ intersects itself infinitely many times then every simple loop of σ surrounds a set of poles whose sum of residues has real part belonging to (−3₂, −1) ∪ (−1, −1₂).

Conjecture 1.3.12 (Abate). Let σ : [0, ε) → So be a maximal geodesic for a meromorphic connection ∇ on P1(C), where So = P1(C) \ {p0, p1, . . . , pr}

and p0, p1, . . . , pr are the poles of ∇. If σ intersects itself infinitely many

times then the ω-limit set of σ is P1(C).

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1.3.2 Local behavior of Fuchsian geodesics

Definition 1.3.13. Let ∇ be a meromorphic connecton on a Riemann sur-face S and p0a Fuchsian pole for ∇. A Fuchsian pole p0 is said to be resonant

if −1 − Resp0∇ ∈ N

∗_{, non-resonant othervise.}

To study the ω-limit set of a geodesic of a meromorphic connection ∇ on a Riemann surface it is useful to know local behavior of geodesics around a pole p0. The next theorem gives such an understanding around the non-resonant

Fuchsian poles.

Theorem 1.3.14 ([8, Proposition 8.4]). Let ∇ be a meromorphic connection on a Riemann surface S. Let p0 be a Fuchsian pole for ∇. Let ρ := Resp0∇.

Suppose −1 − ρ /_{∈ N}∗. Then there is a neighborhood U of p0 such that:

1. if Re ρ < −1 then all geodesics but one issuing from any point p ∈ U \ {p0} tend to p0 staying inside U (the only exception escapes U );

2. if Re ρ > −1 then all geodesics but one issuing from any point p ∈ U \ {p0} escapes U ;

3. if Re ρ = −1 but ρ 6= −1 then the geodesics not escaping U are either closed or accumulate the support of a closed geodesic in U ;

4. if ρ = −1 then any maximal geodesic σ : I → U \ {p0}, maximal in

both forward and backward time, is either periodic or escapes U in one ray and tends to p0 in another ray.

In [8] the resonant case is also studied and the following conjectured is advanced

Conjecture 1.3.15 (Abate-Tovena). Let ∇ be a meromorphic connection on a Riemann surface S. Let p0 be a Fuchsian pole for ∇. Let ρ := Resp0∇.

Suppose −1 − ρ ∈ N∗. Then there is a neighborhood U of p0 such that all

geodesics but one issuing from any point p ∈ U \ {p0} tend to p0 staying

inside U (the only exception escapes U ).

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Chapter 2 Singular flat metrics on

Riemann surfaces

In this chapter we introduce the notion of singular flat metrics on a Rie-mann surface S and we study some properties of singular flat metrics. In Section 2.1, we present general properties of singular flat metrics. In Section 2.2, we describe a relation between singular flat metrics and meromorphic connections.

2.1 Singular flat metric

In this section we define notion of singular flat metrics and we study some of its properties.

Definition 2.1.1. Let S be a Riemann surface and Σ = {p1, ..., pr} a finite

set. Set So := S \ Σ. We say that g is a singular flat metric on S, if g is a flat metric on So _{and for any p ∈ Σ there exist c}

p, bp ∈ R with bp > 0 such

that if (Uα, zα) is a chart centered p, with Uα∩ Σ = {p}, then the flat metric

g12 = euα|dz_α| on U_α\ {p} satisfies lim zα→0 euα |zα|cp = bp

where uα : Uα\{0} → R is a harmonic function. We say that cp is the residue

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Remark 2.1.2. Indeed, the residue cp does not depend on the chosen chart.

Let (Uβ, zβ) be another chart centered p and g

1

2 = euβ|dz

β| on Uα\ {p} for

some harmonic function uβ : Uα\ {p} → R. Then we have

euβ _{= e}uα_|ξ αβ| , where ξαβ = dz_dzα β. Consequently, lim zβ→0 euβ |zβ|cp = lim zβ→0 euα |zβ|cp |ξαβ| = lim zβ→0 euα_|z α|cp |zα|cp|zβ|cp |ξαβ| = bp|ξαβ(p)|cp+1.

Since ξαβ(p) 6= 0, we conclude that cp does not depend on the chosen chart.

If the residue of a critical point p is cp = 0, then it is a removable critical

point i.e., the flat metric g extends (remaining flat) to the critical point. The interesting case is when cp 6= 0.

Lemma 2.1.3. If g is a singular flat metric on a Riemann surface S and p a critical point with the residue cp then for any simply connected chart

(U, z) centered at p with U ∩ Σ = {p} there exists a holomorphic function F : U → C such that the flat metric is given by

g12 = |z|cp|eF (z)dz|

on U \ {p}.

Proof. By definition of meromorphic flat surface there exists bp > 0 such that

lim

z→0

eu |z|cp = bp

where e2u _{is the representation of g on U \ {p}. Let define a function v :}

U \ {p} → R as follows

v(z) := u(z) − cplog|z|.

The function is harmonic on U \ {p} and lim

z→0v(z) = log bp

which means v extends harmonically to U . Since U is simply connected, there exists a holomorphic function F on U such that Re F = v. Then we have

g12 = ev+cplog|z||dz|= |z|cp|eFdz|

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2.1.1 _{Singular flat metrics on C}

Let us study singular flat metrics on a simply connected domain D ⊂ C. Proposition 2.1.4. Let D ⊆ C be a simply connected domain. Let g be a singular flat metric on D. Let Σ := {z1, ..., zn} ⊂ D be the set of singularities

of g and ρj the residue of g at zj for j = 1, ..., n. Then there exists a

holomorphic function F : D → C such that

g12 = |z − z₁|ρ1·|z − z₂|ρ2... · |z − z_n|ρnexp(Re F )|dz|.

Proof. Set Do _{= D \ Σ. Then there is a harmonic function u : D}o _{→ R such}

that

g12 = exp(u)|dz|.

Set

v = u − ρ1log|z − z1|−ρ2log|z − z2|−... − ρnlog|z − zn|

It is easy to see that v is harmonic on Do. Similarly as Lemma 2.1.3 we can show that v has harmonic continuation to any critical point zj, for j = 1, .., n.

Hence v is a harmonic function on D. Since D is simply connected, there exists a holomorphic function F : D → C such that Re F = v. Consequently,

g12 = |z − z₁|ρ1·|z − z₂|ρ2... · |z − z_n|ρnexp(Re F )|dz|.

Definition 2.1.5. Let

g12 = |z − z₁|ρ1·|z − z₂|ρ2... · |z − z_n|ρnexp(Re F )|dz|= u(z)|dz|

be a singular flat metric on a simply connected domain D ⊂ C, where F : D → C is a holomorphic function and ρ1, ρ2, ..., ρn the residues of g at the

singularities z1, ..., zn respectively. We say u is the representation of the

singular flat metric g on D.

Some properties of singular flat metrics of the form |f |λ_{|dz| on a simply}

connected domain D ⊆ C are described in [38], where λ > 0 and f : D → C is a holomorphic function.

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2.2 Relation between singular flat metrics and

meromorphic connections

In this section we describe a relation between singular flat metrics and meromorphic connections with real periods. Let ∇ be a holomorphic connec-tion on a Riemann surface S. As we have seen in Theorem 1.1.7 there exists a flat (Proposition 1.1.6) metric g adapted to ∇ if and only if ∇ has real periods. Note that if g is adapted to ∇ then they have the same geodesics. Definition 2.2.1. Let ∇ be a meromorphic connection on a Riemann surface S and let Σ denote the set of poles of ∇. We say that a singular flat metric g on S is adapted to ∇ if it is adapted to ∇ on So := S \ Σ.

Definition 2.2.2. Let S be a Riemann surface and Σ ⊂ S a discrete set not having limit point in S. Set So = S \ Σ. A Leray atlas adapted to (So, Σ) is an atlas {(Uα, zα)} ∪ {(Uk, zk)}, where {(Uα, zα)} is a Leray atlas for So,

each (Uk, zk) is a simply connected chart centered at pk∈ Σ and Uk∩ Uh = ∅

if k 6= h.

Let us state an analogue of Theorem 1.1.7

Theorem 2.2.3. Let ∇ be a Fuchsian meromorphic connection on a Rie-mann surface S, and Σ the set of poles of ∇. Set So _{= S \ Σ. If ∇ has}

real periods on So _{and Res∇ ⊂ R then there exists a singular flat metric g} adapted to ∇. Moreover, g is unique up to a positive constant multiple.

Conversely, if g is a singular flat metric on S with singular set Σ then there exists a unique meromorphic connection ∇ with Σ as set of poles such that g is adapted to ∇. Moreover, ∇ has real periods on So _{and Res∇ ⊂ R.} Furthermore, if cp is the residue of a critical point p of g then Resp∇ = cp

and vice versa.

Proof. Let g be a singular flat metric on S and Σ its singular set. Set So _{= S \ Σ. Let {(U}

α, zα)} ∪ {(Uk, zk)} be a Leray atlas adapted to (So, Σ).

Since g is flat on So for each α there exists a harmonic function uα on Uα

such that g

1 2

α = euα|dzα| on Uα. Furthermore, one has

euα−uβ _{= |ξ}

αβ| (2.1)

on Uα ∩ Uβ, where ξαβ = dz_dzα_β. Since Uα is simply connected, there exists

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holomorphic function on Uα. Set

ηα := dfα (2.2)

on Uα; we claim that the ηα’s are representatives of a holomorphic connection

∇ on So_{. By (2.1) we have}

uα− uβ = log|ξαβ|.

Since ξαβ : Uα∩ Uβ → C is a nonzero holomorphic function then

F := fα− fβ− log ξαβ

is a holomorphic function on Uα ∩ Uβ with Re F = 0. Hence there exists

C ∈ R such that F = iC, i.e.,

fα− fβ = log ξαβ + iC

After differentiating the last equality we have ηα− ηβ =

dξαβ

ξαβ

which is (1.2). We have defined a holomorphic connection ∇ on So. It is not difficult to see that g is adapted to ∇ on So_{. By Theorem 1.1.7 it follows}

that ∇ has real periods on So_.

Let pk be a critical point for g with residue ck and (Uk, zk) the chart of

the atlas centered at p. Then by Lemma 2.1.3 there exists a holomorphic function Fk on Uk such that the singular flat metric is

g12 = |z_k|ck|eFkdz_k| on Uk. Set ηk:= ck zk + F_k0 dzk

on Uk. We claim that we can extend ∇ to a meromorphic connection

repre-sented by ηk on Uk. We have to check that this definition satisfies condition

(1.2). Let (Uα, zα) be a chart with Uα ∩ Uk 6= ∅. Then there exists a

holo-morphic map Fα : Uα → C such that

g12 = |eFαdz

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Let V ⊆ Uα∩ Uk be a simply connected open set. Then by definition of flat

metric we have

|zk|ck|eFk|= |eFαξαk|

on V , where ξαk := dz_dzα

k. This is equivalent to

cplog|zk|+Re Fk = Re Fα+ log|ξαk|.

Since V is simply connected it is not difficult to see that there exists a constant C ∈ C such that

cklog zk+ Fβ − Fk− log ξαk = C on V . Consequently, ηk= ηα+ dξαk ξαk (2.3) on V . Since ηk, ηα and ξαk are well defined on Uα ∩ Uk and the equality

(2.3) holds on any simply connected subset of Uα ∩ Uk then (2.3) holds on

Uα∩ Uk. Hence we here extended ∇ to a Fuchsian meromorphic connection

on S. It is easy to see that, if pk is a critical point of g with residue ck then

Respk∇ = ck.

Let ˜∇ be another meromorphic connection adapted to g. Let ηα and ˜ηα

be the representations of ∇ and ˜∇ on a chart (Uα, zα) respectively. Then by

(2.2) there exists holomorphic primitives Fα and ˜Fαof ηα and ˜ηαrespectively

we have

g12 = |eFαdz_α|= |eF˜αdz_α|

on Uα. Hence there exists a constant C ∈ C such that Fα ≡ ˜Fα + C.

Consequently, ηα ≡ ˜ηα. Hence ∇ = ˜∇ on So. Let now (Uk, zk) be a chart

around a pole pk. Let ηk and ˜ηk be the representations of ∇ and ˜∇ on Uk

respectively. Since ∇ = ˜∇ on So _{we have η}

k ≡ ˜ηk on Uk\ {pk}. Since ηk has

unique extension to Uk we have ηk ≡ ˜ηk on Uk. Thus ∇ = ˜∇ on S. Hence ∇

is the unique meromorphic connection adapted to g.

On the other hand, let ∇ be a Fuchsian meromorphic connection on a Riemann surface S with real periods and Res∇ ⊂ R. Let Σ be the set of poles of ∇. Set S \ Σ. By Theorem 1.1.7 there exists a flat metric g adapted to ∇ on So. Let {(Uα, zα)} ∪ {(Uk, zk)} be a Leray atlas adapted to (So, Σ).

Let ηα be the representation of ∇ on Uα. By Proposition 1.1.6 g is defined

g

1 2

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on Uα for a suitable holomorphic primitive Fα of ηα. Let pk be a pole of ∇.

Let (Uk, zk) be the chart centered at p. Let

ηk= c_k zk + fk dzk

be the representation of ∇ on Uk, where fk : Uk → C is a holomorphic

function and ck := Respk∇. Let V ⊂ Uk\ {pk} be a simply connected open

set. Then by Proposition 2.2 for a suitable holomorphic primitive Kk of ηk

on V we have

g12 = exp(Re K

k)|dzk|.

Then

Kk = cklog zk+ Fk

for a holomorphic primitive Fk of fk on V . Hence

g12 = |z_k|ck|eFkdz_k|

on V . Since Fk is a holomorphic primitive of fkon V and fkis a holomorphic

function on Uk there exists a holomorphic primitive ˜Fk of fk on Uk such that

˜ Fk|V= Fk. Consequently, we have g12 = |z_k|ck|eF˜kdz_k| on Uk. Hence lim zk→0 |zk|ck|e ˜ Fk_| |zk|ck = eF (0)˜ .

By definition of singular flat metric we can see that pk is a singular point for

g with residue ck. Hence g has a continuation as a singular flat metric to any

pole of ∇. Since g is adapted to ∇ on So _{it is adapted to ∇ on S.}

Let ˜g be another singular flat metric adapted to ∇. Let ηα be the

repre-sentation of ∇ on (Uα, zα). For suitable holomorphic primitives Fα and ˜Fα

of ηα we have g 1 2 α = exp(Re Fα)|dzα| and ˜ g 1 2 α = exp(Re ˜Fα)|dzα|

on Uα. Since Fα and ˜Fα holomorphic primitives of ηα there exists Cα ∈ C

such that

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Hence g 1 2 α = |eCα|˜g 1 2

α on Uα. Let (Uβ, zβ) be a chart with Uα∩ Uβ 6= ∅. Then

we have g 1 2 α = g 1 2 β on Uα∩ Uβ, and it is equivalent to |eCα_|˜_g 1 2 α = |eCβ|˜g 1 2 β.

Hence |eCα_{|= |e}Cβ|= r for some r > 0. Consequently, we have g = r2_g._˜

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Chapter 3 Meromorphic quadratic

differentials

Meromorphic quadratic differentials are a well studied subject in Te-ichm¨uller theory. Studying quadratic differentials is a way to understand interval exchange transformations (see for example [19, 38, 45, 46]). The main purpose of this chapter is to describe the relation between meromor-phic k-differentials and meromormeromor-phic connections.

In Section 3.1, we recall some definitions and fundamental results on the theory of meromorphic quadratic differentials and we review a relation between meromorphic quadratic differentials and singular flat metrics. In Section 3.2, we recall notion of argument of a quadratic differential along a smooth curve σ. Furthermore, its contents includes some fundamental results like uniquely ergodicity of a quadratic differentials on a compact Rie-mann surface, and relation between interval exchange transformations and quadratic differentials. Section 3.3 covers material about meromorphic k-differentials. In Section 3.4 we describe a relation between meromorphic connections and meromorphic k-differentials.

3.1 Quadratic differentials

Let us begin recalling a few standard facts about meromorphic quadratic differentials on a Riemann surface S (see, e.g., [38]).

Definition 3.1.1. Let {(Uα, zα)} be a holomorphic atlas for a Riemann

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function elements qα defined in local charts (Uα, zα) for which the following

transformation law holds: qβ(p) = dz_α dzβ (p) 2 qα(p), (3.1)

for any p ∈ Uα∩ Uβ. Usually, we say qα is the local representation of q on

Uα.

Globally, a quadratic differential is a global meromorphic section of the line bundle (T∗S)⊗2.

It makes no sense to speak about the value of a quadratic differential q at a point p ∈ S, since it depends on the local chart near p, but we can speak of its zeros and poles.

Definition 3.1.2. Let q be a meromorphic quadratic differential on a Rie-mann surface S and p a zero (pole) of q. The order of p is the order of zα(p)

for any qα representing q on a local chart (Uα, zα) around p.

Indeed, dzα

dzβ is a never vanishing holomorphic function on Uα∩ Uβ, and

hence (3.1) yields that qα and qβ have the same order of zero (respectively,

pole) at p.

Definition 3.1.3. The critical points of a meromorphic quadratic differential q on a Riemann surface S are its zeroes and poles. All other points of S are regular points of q. A finite critical point of q is either a zero or a pole of order one. Other critical points will be called infinite critical points. If q has no poles then we say that q is a holomorphic quadratic differential. If all points are regular points for q then we say that q is a regular quadratic differential.

Proposition 3.1.4 ([38, Theorem 5.1]). Let q be a regular quadratic differ-ential on a Riemann surface S. Then there exists an atlas {(Uα, zα)} for S

such that the representation of q is identically one on each Uα. The atlas

have transition functions zβ = ±zα+ cαβ on Uα∩ Uβ.

Hence if q is a meromorphic quadratic differential on a Riemann surface S and p is a regular point for q then there exists a chart around p such that the representation of q on this chart is identically one. Of course this is impossible around critical points of q. Next theorem gives an understanding of local behavior around critical points

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Theorem 3.1.5 ([38, Chapter 3]). Let p0 be a critical point of a meromorphic

quadratic differential q of order n. Then there is a chart (U, z) centered p0,

such that

1. if n ∈ N or is a negative odd number then q = n + 2

2 2

zndz2

on U ;

2. if n = −2 then there exists a ∈ C∗ depending only the meromorphic quadratic differential q such that

q = a

z2dz 2

on U ;

3. if n ≤ −4 is an even number then there is b ∈ C depending only the meromorphic quadratic differential q such that

q = n + 2 2 z n 2 + b z 2 dz2 on U .

3.1.1 The metric associated with a quadratic

differen-tial

In this subsection we study relation between singular flat metrics and quadratic differentials.

Proposition 3.1.6 ([38, Section 5.3]). Let q be a meromorphic quadratic differential on a Riemann surface S and {(Uα, zα)} an atlas for S. Then

there exists a singular flat metric g on S locally given by g12 = |q_α|

1

2|dz_α| (3.2)

on Uα, where qα is the local representation of q on Uα.

In the end of the chapter we will prove the converse of the previous proposition.

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Definition 3.1.7. We say the singular flat metric (3.2) is adapted to the quadratic differential q.

Corollary 3.1.8. Let q be a meromorphic quadratic differential on a Rie-mann surface S. Then there exists a unique singular flat metric adapted to q.

Proof. It follows by (3.2).

Let us define the notion of geodesic for a meromorphic quadratic differ-ential q on a Riemann surface S.

Definition 3.1.9. Let q be a meromorphic quadratic differential on a Rie-mann surface S and g is the singular flat metric adapted to q. Set So _{:= S \Σ,}

where Σ is the set of critical points of q. A smooth curve σ : [0, ε) → So _{is a}

geodesic for q if it is a geodesic for g.

Proposition 3.1.10 ([38, Theorem 5.5]). Let q be a meromorphic quadratic differential on a Riemann surface S. Then there are no self-intersecting geodesics of q except periodic geodesics.

The next theorem describes a local behavior of geodesics of a quadratic differential q around its critical points.

Theorem 3.1.11 ([38, Chapter 3]). Let q be a meromorphic quadratic dif-ferential on a Riemann surface S. Let p0 be a critical point for q with order

k. Then there is a neighborhood U of p0 such that:

1. if k < −2 then all geodesics but one issuing from any point p ∈ U \ {p0}

tend to p0 staying inside U (the only exception escapes U );

2. if k ≥ −1 then all geodesics but one issuing from any point p ∈ U \ {p0}

escape U ;

3. if k = −2 then any maximal geodesic σ : I → U \ {p0}, maximal in

both forward and backward time, is either periodic or escapes U in one ray and tends to p0 in the other ray.

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3.2 Horizontal foliation and unique

ergodic-ity

Let q be a meromorphic quadratic differential on a Riemann surface S and {(Uα, zα)} an atlas for S. Let σ : [0, ε) → Uα be a smooth curve. Take

the pull-back

qα(σ(t))(σα0(t)) 2_dt2

of qαdzα2, where σα= zα(σ). Then we define

arg_q_ασ(t) := arg qα(σ(t))(σ0α(t)) 2_.

Suppose (Uβ, zβ) is another chart such that Uα ∩ Uβ∩ suppσ 6= ∅. Then by

transformation rule (3.1) we have

arg_q_ασ(t) = arg_q β(σ(t)) on Uα∩ Uβ, because σ0_β(t) = dzβ dzα (σ(t))σ_α0(t).

Hence arg_qσ(t) is globally defined.

Remark 3.2.1. By definition we can see that arg_qσ(t) does not depend on the parametrization of σ.

Theorem 3.2.2. [38] Let q be a meromorphic quadratic differential on a Rieamann surface S. Set So _{:= S \ Σ, where Σ is the set of critical points of}

q. Then a smooth curve σ : [0, ε) → So is a geodesic for q if and only if

arg_qσ(t) = θ = const

for some θ ∈ [0, 2π).

Definition 3.2.3. Let q be a meromorphic quadratic differential on a Riea-mann surface S. Set So _{:= S \ Σ, where Σ is the set of critical points of q.}

Let θ ∈ [0, 2π). We say that a geodesic σ : [0, ε) → So _{of q has θ-trajectory}

if arg_qσ(t) ≡ θ. We say σ is horizontal if θ = 0 and vertical if θ = π.

Remark 3.2.4. Let So be the complement of the critical set of q. Note that q is locally square of an Abelian differential on So_{. Therefore σ is vertical}

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Note that if σ has θ−trajectory then σ is a horizontal geodesic for e−iθq. In particular vertical geodesics of q are the horizontal geodesics for −q. It is therefore sufficient to study horizontal geodesics of quadratic differentials. Theorem 3.2.5 ([38, Theorem 5.5]). Let q be a meromorphic quadratic dif-ferential on an arbitrary Riemann surface S. Then through every regular point of q there exists a uniquely determined horizontal geodesic.

In particular, two horizontal geodesics never have a common point unless they coincide.

Definition 3.2.6. Let q be a quadratic differential on a compact Riemann surface S. The horizontal foliation induced by the quadratic differential q is a (singular) foliation on the surface whose leaves are the horizontal geodesics maximal in both backward and forward time.

Definition 3.2.7. Let q be a holomorphic quadratic differential on a compact Riemann surface S. We say that a Borel probability measure µ on S is ergodic for q if for every µ-measurable set E ⊆ S which is a union of leaves of the horizontal foliation, µ(E) = 0 or µ(E) = 1. We say q is uniquely ergodic if there is precisely one ergodic measure for q.

An important result about uniquely ergodicity of a holomorphic quadratic differential q on a compact Riemann surface S is given by Kerckhoff, Masur, and Smillie (see [26]).

Theorem 3.2.8 ([26, Theorem 2]). Let q be a holomorphic quadratic differ-ential on a compact Riemann surface S. Then for almost all θ ∈ [0, 2π) the quadratic differential eiθq is uniquely ergodic.

3.2.1 Interval exchange transformation

The class of interval exchange transformations is a classical topic in Dy-namics. They have been first introduced by V. Oseledets [33], and then have been extensively studied (see, e.g., [33, 45, 46, 26]). Let us recall definition of an interval exchange transformation.

Definition 3.2.9. Let λ := (λ1, λ2, ..., λn) ∈ Rn be such that λj > 0 for

j = 1, ..., n. Set β0 = 0, βi := i X j=1 λj

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and Ii := [βi−1, βi). Let π be a permutation of {1, ..., n}. Set

λπ = (λπ−1₍₁₎, ...λ_π−1_(n)).

Form the corresponding βπ

i (λπ) and Iiπ. We define a map τ from Iλ = [0, βn)

to itself by

τ (x) := x − βi−1+ βπ(i)−1π ,

for x ∈ Ii and 1 ≤ i ≤ n. Then τ maps each Ii isometrically onto I_π(i)π and is

called interval exchange transformation.

It is obvious from the definition that τ preserves Lebesgue measure on [0, βn).

Definition 3.2.10. We say that an interval exchange transformation τ is uniquely ergodic if every finite invariant measure is a multiple of the Lebesgue measure.

Interval exchange transformations have a strong relation with holomor-phic quadratic differentials on compact Riemann surfaces.

Theorem 3.2.11 ([38, Section 12.4]). Let q be a holomorphic quadratic dif-ferential on a compact Riemann surface S. Then there exists an interval exchange transformation τ : [0, 1) → [0, 1) such that q is uniquely ergodic if and only if τ is uniquely ergodic.

On the other hand, if τ : [0, 1) → [0, 1) is an interval exchange transfor-mation then there exists a holomorphic quadratic differential q on a compact Riemann surface S such that τ is uniquely ergodic if and only if q is uniquely ergodic.

3.3 Meromorphic k-differentials

Meromorphic k-differential is studied by many authors (see for example [10, 36, 40, 41]). In this section we recall some results on the theory of meromorphic k-differentials.

Definition 3.3.1. Let k ∈ N. Let {(Uα, zα)} be a holomorphic atlas for a

Riemann surface S. A meromorphic k-differential q on S is a set of mero-morphic function elements qα defined in local charts (Uα, zα) for which the

following transformation law holds: qβ(p) = dzα dzβ (p) k qα(p), p ∈ Uα∩ Uβ (3.3)

Dynamics of geodesics for meromorphic connections on Riemann surfaces