Meromorphic Extensions of Green's Functions on a Riemann Surface
aa r X i v : . [ m a t h . C V ] D ec MEROMORPHIC EXTENSIONS OF GREEN’S FUNCTIONS ON ARIEMANN SURFACE
MICHAEL P. TUITE
Abstract.
For a Riemann surface of genus g ě G N p x, y q which transforms as a weight N ě x and a weight 1 ´ N formin y and is meromorphic in x , with a unique simple pole at x “ y , but is not mero-morphic in y . For a Schottky uniformized Riemann surface we consider meromorphicextensions of G N p x, y q called Green’s Functions with Extended Meromorphicity orGEM forms. GEM forms are meromorphic in both x and y with a unique simple poleat x “ y , transform as weight N ě x but as weight 1 ´ N quasiperiodicforms in y . We give a reformulation of the bijective Bers map and describe a choiceof GEM form with an associated canonical basis of normalized holomorphic N -forms.We describe an explicit differential operator constructed from N “ Introduction
For every Riemann surface of genus g ě G N p x, y q with the following properties [Ma, McIT]: G N p x, y q transforms as a weight N ě x , as a weight 1 ´ N differential form in y and is meromorphicin x with a unique simple pole at x “ y but is not meromorphic in y where BB y G N p x, y q has a specific form. Using a Schottky uniformization of the Riemann surface, thedefinition of G N p x, y q utilizes a Poincar´e series Ψ Bers N p x, y q introduced by Bers [Be1, Be2]as a means of constructing a Bers potential for each holomorphic weight N differentialform. Unlike the Green’s function, Ψ Bers N p x, y q is meromorphic in both x and y witha unique simple pole at x “ y and transforms as a weight N differential form in x but transforms as a quasiperiodic differential form of weight 1 ´ N in y . Ψ Bers N p x, y q is an example of what we call a Green’s Function with Extended Meromorphicity orGEM form, denoted by Ψ N p x, y q . GEM forms can be viewed as a generalization of theclassical differential of the third kind ω y ´ p x q which is meromorphic in both x and y andis a weight 1 differential form in x and a quasiperiodic differential form of weight 0 in y but unlike a GEM form, possesses two simple poles at x “ y and x “ N -differentials which generalizes the canonical basisof holomorphic 1-differentials associated with ω y ´ p x q . In Section 3 we discuss the ge-ometrical meaning of N “ g ´ space of holomorphic quadratic differentials. In particular, employing a canonical pa-rameterization of Schottky space, we explicitly construct a new canonical differentialoperator (partially anticipated in ref. [O]) employing Ψ p x, y q which gives the variationwith respect to moduli space parameters of a punctured Riemann surface. In Section 4the general relationship between Ψ N p x, y q and the Green’s function G N p x, y q is furtherdeveloped and we describe the inverse Bers map giving the Bers potential for a givenchoice of Eichler cocycle by means of a new expression involving G N p x, y q . Acknowledgements.
I wish to thank Michael Flattery, Tom Gilroy and MichaelWelby for helpful comments and suggestions.2.
The Bers Map and GEM forms
The Schottky uniformization of a Riemann surface.
We briefly review theconstruction of a genus g Riemann surface S g using the Schottky uniformization wherewe sew g handles to the Riemann sphere S – p C : “ C Y8 e.g. [Fo, Bo]. Every Riemannsurface can be Schottky uniformized [Be3]. Let t C ˘ a u , where a P I ` “ t , . . . , g u , denotea set of 2 g non-intersecting Jordan curves in C . Identify z P C ´ a with z P C a via theSchottky sewing cross ratio relation z ´ W ´ a z ´ W a z ´ W a z ´ W ´ a “ q a , a P I ` , (1)for complex q a with 0 ă | q a | ă W ˘ a P p C . Thus z “ γ a z for a P I ` forM¨obius transformation generated by γ a P SL p C q where γ a : “ σ ´ a ˜ q { a q ´ { a ¸ σ a , σ a “ p W ´ a ´ W a q ´ { ˆ ´ W ´ a ´ W a ˙ . (2) γ a is loxodromic with attracting fixed point W ´ a and repelling fixed point W a .The marked Schottky group Γ Ă SL p C q is the free discrete group of M¨obius trans-formations generated by γ a . Let Λ p Γ q denote the limit set. Then S g » Ω p Γ q{ Γ, aRiemann surface of genus g . We let D Ă p C denote the standard connected fundamentalregion with oriented boundary curves C a . We further identify the standard homologycycle α a with C ´ a and the cycle β a with a path connecting z P C a to z “ γ a z P C ´ a .Define γ ´ a “ γ ´ a so that γ a C a “ ´ C ´ a for all a P I “ t˘ , . . . , ˘ g u . Let w a : “ γ ´ a . so that w a ´ W a w a ´ W ´ a “ q a which implies w a “ W a ´ q a W ´ a ´ q a , (3)for all a P I . We therefore have γ a z “ w ´ a ` ρ a z ´ w a , (4)where ρ a “ ρ ´ a is determined from the condition γ a W a “ W a to be ρ a “ ´ q a p W a ´ W ´ a q p ´ q a q . (5)Hence (1) can be written in the more convenient form: p z ´ w ´ a qp z ´ w a q “ ρ a . (6) EROMORPHIC EXTENSIONS OF GREEN’S FUNCTIONS ON A RIEMANN SURFACE 3
We may choose the Jordan curve C a to be the isometric circle of γ a of radius | ρ a | centred at w a . We note that the interior/exterior of the disk ∆ a “ t z : | z | ď | ρ a | u is mapped by γ a to the exterior/interior of ∆ ´ a since | γ a z ´ w ´ a || z ´ w a | “ | ρ a | .Furthermore, the fixed point W a P ∆ a . ✫✪✬✩ ¨ w a ¨ W a ✲ C a ✫✪✬✩✫✪✬✩ ¨ w ´ a ¨ W ´ a ✲ C ´ a ✲ γ a Fig. 1 Isometric Schottky Circles
We define the space of Schottky parameters C g Ă C g by C g : “ ! p w , w ´ , ρ , . . . , w g , w ´ g , ρ g q : | w a ´ w b | ą | ρ a | ` | ρ b | @ a ‰ b ) , (7)where the condition follows from ∆ a X ∆ b “ H for a ‰ b . The cross ratio (1) is M¨obiusinvariant with p z, z , W a , q a q ÞÑ p γz, γz , γW a , q a q for γ “ p A BC D q P SL p C q giving thefollowing SL p C q action on C g γ : p w a , ρ a q ÞÑ ˆ p Aw a ` B q p Cw ´ a ` D q ´ ρ a AC p Cw a ` D q p Cw ´ a ` D q ´ ρ a C , ρ a pp Cw a ` D q p Cw ´ a ` D q ´ ρ a C q ˙ . (8)We define Schottky space as S g “ C g { SL p C q . S g is a covering space for the 3 g ´ M g of genus g Riemann surfaces e.g. [Be3].2.2.
Holomorphic differentials H p g q N . Let A m,n for m, n P Z denote the vector spaceof smooth differentials of the form Φ p z q “ φ p z q dz m dz n for local coordinate z on S g e.g.the Poincar´e metric R p z q “ ρ p z q dzdz P A , for real positive ρ p z q . R determines thepositive definite Petersson inner product for Φ , Θ P A m,n x Φ , Θ y : “ ij S g Φ Θ R ´ m ´ n ω, (9)for real volume form ω p z q : “ ρ p z q d z, (10)with d z : “ i2 dz ^ dz . Let H p g q m,n denote the L -closure of A m,n with respect to thePetersson product and let H p g q n “ H p g q n, . Lastly, let H p g q N Ă H p g q N denote the space of genus g holomorphic N -differentials where N is referred to as the weight. The Riemann-Rochtheorem (e.g. [FK, Bo]) determines d N “ dim H p g q N as follows: The isometric circle for ` a bc d ˘ P SL p C q is given by | cz ` d | “ R is induced from the Poincar´e metric y ´ dζdζ on H “ t ζ “ x ` i y | x, y P R , y ą u by uniformizing S g as a quotient of H by an appropriate Fuchsian subgroup of SL p R q . MICHAEL P. TUITE genus g weight N dimension d N g “ N ď ´ NN ą g “ N P Z g ě N ă N “ N “ gN ě p g ´ qp N ´ q In the Schottky uniformization, Φ p z q “ φ p z q dz m dz n P A m,n for z P Ω p Γ q satisfiesΦ | γ “ Φ , for all γ P Γ where Φ | γ p z q : “ φ p γz q d p γz q m d p γz q n and the Petersson product is expressedas an integral over the Schottky fundamental region D e.g. [McI, McIT].2.3. Bers potentials and GEM forms.
We review the relationship between H p q ´ N and H p g q N in the Schottky scheme for all N ě g ě k denote the k ` ď k . H p q ´ N consists of elements P p z q “ p p z q dz ´ N for z P p C and p P Π N ´ .There is a natural M¨obius action on H p q ´ N given by P | γ p z q : “ P p γz q , (11)for γ P SL p C q with P | γλ “ P | γ | λ for all γ, λ P SL p C q .Let Γ be a Schottky group for a Riemann surface of genus g ě
2. Let Z p Γ , H p q ´ N q denote the vector space of Eichler 1-cocycles for Γ given by mappings of the formΞ : Γ Ñ H p q ´ N such that for all γ, λ P ΓΞ r γλ s “ Ξ r γ s| λ ` Ξ r λ s , (12)for M¨obius action (11). We note that (12) implies Ξ r id s “ r γ s “ ´ Ξ r γ ´ s| γ .Let B p Γ , H p q ´ N q Ă Z p Γ , H p q ´ N q denote the space of coboundaries Ξ P : Γ Ñ H p q ´ N for P P H p q ´ N given by Ξ P r γ s : “ P | γ ´ P. (13)Ξ P is a 1-cocycle since P | γ | λ ´ P “ p P | γ ´ P q | λ ` P | λ ´ P . It is easy to show that [Be1] Lemma 2.1. B p Γ , H p q ´ N q » H p q ´ N as vector spaces. Let H p Γ , H p q ´ N q : “ Z p Γ , H p q ´ N q{ B p Γ , H p q ´ N q be the cohomology space of Eichlercocycles modulo coboundaries. Lemma 2.2. [Be1] dim H p Γ , H p q ´ N q “ p g ´ qp N ´ q .Proof. Since Γ is freely generated by t γ a u for a P I ` , a cocycle Ξ is determined by itsevaluation on γ a . Thusdim Z p Γ , H p q ´ N q “ g dim H p q ´ N “ g p N ´ q . The result follows on applying Lemma 2.1. (cid:3) We define a 1-cocycle as a mapping to H p q ´ N rather than to Π N ´ as in refs. [Be1, G]. EROMORPHIC EXTENSIONS OF GREEN’S FUNCTIONS ON A RIEMANN SURFACE 5
Note that dim H p Γ , H p q ´ N q “ dim H p g q N for N ě g ě
2. The Bers map discussedbelow describes a bijection between these spaces. F p y q “ f p y q dy ´ N for f p y q continuous for y P Ω p Γ q is called a Bers potential for aholomorphic N -form Φ “ φ p y q dy N P H p g q N provided f p y q satisfies π B y f “ φ p y q ρ p y q ´ N , (14) lim y Ñ ˇˇ y N ´ f ` y ´ ˘ˇˇ ă 8 , (15)where B y “ BB y . (14) can also be written in the following coordinate-free way12 π i d p F Θ q “ Θ Φ R ´ N ω, (16)for all holomorphic Θ P H p g q N with exterior derivative d p h p y q dy q “ ´B y h dy ^ dy andvolume form ω of (10). (15) ensures that F p y q is defined at the point at infinity. Let F N denote the vector space of Bers potentials. It is straightforward to see that Lemma 2.3. [Be1] F P F N is a Bers potential for Φ “ iff F P H p q ´ N . A Bers potential F is quasiperiodic under the action of the Schottky group. DefineΞ F r γ s : “ F | γ ´ F, γ P Γ , (17)where F | γ p y q “ F p γy q . We then find Lemma 2.4. Ξ F is an Eichler 1-cocycle for each F P F N .Proof. (16) implies d pp F Θ q| γ ´ F Θ q “ P H p g q N so that Ξ F r γ s P H p q ´ N . Ξ F isa 1-cocycle since F | γλ ´ F “ p F | γ ´ F q | λ ` F | λ ´ F . (cid:3) The existence of a Bers potential for Φ P H p g q N for N ě g ě x, y P Ω p Γ q by [Be1, Be2]Ψ Bers N p x, y q : “ ÿ γ P Γ γx ´ y N ´ ź j “ y ´ A j γx ´ A j d p γx q N dy ´ N . (18)Here A , . . . , A N ´ are any distinct elements of the limit set Λ p Γ q . Ψ Bers N p x, y q is mero-morphic in x, y P Ω p Γ q with a simple pole of residue one at y “ γx for all γ P Γ[Be1, Be2, G, McI, McIT]. Furthermore, Ψ
Bers N p x, y q is a bidifferential p N, ´ N q -quasiform with respect to the Schottky group as follows. By construction, Ψ Bers N p x, y q is an N -differential in x so that for all γ P ΓΨ Bers N p γx, y q “ Ψ Bers N p x, y q . (19)Ψ Bers N p x, y q is a quasiperiodic 1 ´ N form in y whereΨ Bers N p x, γy q ´ Ψ Bers N p x, y q “ χ Bers r γ sp x, y q , (20)for γ P Γ where χ Bers r γ sp x, y q is holomorphic for x, y P Ω p Γ q .For a given N -differential Φ P H p g q N and y P Ω p Γ q we define F Bers p y q : “ ´ ij D Ψ Bers N p¨ , y q Φ R ´ N ω “ ´x Ψ Bers N p¨ , y q , Φ y . We include a factor of π in comparison to [Be1, McIT] for later convenience. MICHAEL P. TUITE F Bers p y q “ f Bers p y q dy ´ N satisfies (15) with f Bers p y q continuous on Ω p Γ q [Be1]. Since f p y q “ ´ π B y ij R f p z q z ´ y d z, y P R, (21)for any complex function f p z q on an open region R Ă C (e.g. [GR]), we find that F Bers p z q satisfies (14). Thus F Bers p y q is a Bers potential for Φ with cocycle Ξ Bers from(17) and we find:
Proposition 2.1. [Be1]
There exists a Bers potential for each Φ P H p g q N . By Lemma 2.3 the most general Bers potential for Φ P H p g q N is of the form F “ F Bers ` P for some P P H p q ´ N with 1-cocycle Ξ “ Ξ Bers ` Ξ P . Let t Φ r u d N r “ be a H p g q N -basisof dimension d N “ p g ´ qp N ´ q where Φ r has potential F r p y q “ F Bers r p y q ` P r p y q withcocycle Ξ r “ Ξ Bers r ` Ξ P r for some choice of P r P H p q ´ N . Let t Φ _ r u d N r “ be the Peterssondual basis. We may define the following meromorphic bidifferential p N, ´ N q -quasiformΨ N p x, y q : “ Ψ Bers N p x, y q ´ d N ÿ r “ Φ _ r p x q P r p y q , (22)where the Bers potential for Φ r is given by F r p y q “ ´x Ψ N p¨ , y q , Φ r y . (23)Ψ N p x, y q is an N -form in x and a quasiperiodic 1 ´ N form in y withΨ N p x, γy q ´ Ψ N p x, y q “ χ r γ sp x, y q , (24)where χ r γ sp x, y q is holomorphic for all x, y P Ω p Γ q . Hence (19) and (23) imply χ r γ sp x, y q “ ´ d N ÿ r “ Φ _ r p x q Ξ r r γ sp y q . (25)We refer to Ψ N of (22) as a Green’s function with Extended Meromorphicity or aGEM form for reasons explained in §
4. The space of GEM forms is of dimension p N ´ q d N “ p g ´ qp N ´ q from (22).2.4. The Bers map.
The following commutative diagram summarizes the variousmaps introduced in the previous section F N α
ÝÝÝÑ Z p Γ , H p q ´ N q ǫ §§đ δ §§đ H p g q N β
ÝÝÝÑ H p Γ , H p q ´ N q (26)where α is the linear map determined by (17), ǫ is the complex anti-linear map deter-mined by (14) with pre-image determined by (23) and δ is the coboundary quotientmap with B p Γ , H p q ´ N q “ ker δ . Then Lemmas 2.1 and 2.3 are equivalent toker ǫ “ H p q ´ N , ker δ “ α p ker ǫ q . (27)The complex antilinear mapping β is known as the Bers map. We have the followingfundamental result (which is a reformulation of Bers’ classic result [Be1]) Proposition 2.2. F N » Z p Γ , H p q ´ N q as vector spaces. EROMORPHIC EXTENSIONS OF GREEN’S FUNCTIONS ON A RIEMANN SURFACE 7
Proof.
We first show that α : F N Ñ Z p Γ , H p q ´ N q is injective. Let F P F N be a potentialfor Φ P H p g q N such that Ξ F “ α p F q “
0. Thus F | γ “ F for all γ P Γ implying F P H p g q ´ N .Hence (16) implies that for all Θ P H p g q N we have x Θ , Φ y “ π i ij D d p F Θ q “ ´ ÿ a P I π i ¿ C a Θ F “ ´ π i g ÿ a “ ¿ C a Θ p F ´ F | γ a q “ , (28)by Stokes’ theorem on the fundamental domain D formed from p C by excising 2 g discswith oriented boundary curves C a and that C ´ a “ ´ γ a C a . Since x , y is invertible we haveΦ “ F is holomorphic by (16). Hence F P H p g q ´ N so that F “ F N “ g p N ´ q “ dim Z p Γ , H p q ´ N q so that α is also surjective and thus bijective. (cid:3) Remark 2.1.
Proposition 2.2 together with (27) imply that the Bers map β : H p g q N Ñ H p Γ , H p q ´ N q is also bijective. The statement and proof of Proposition 2.2 can beadapted to any Kleinian group Γ since D consists of a finite set of disconnected com-ponents [Be1] and hence x Θ , Φ y “
0, as in (28), so that α and β are injective.We note the following useful identity [Be1, McIT] Proposition 2.3.
Let Ξ P Z p Γ , H p q ´ N q with Φ “ p ǫ ˝ α ´ q p Ξ q P H p g q N . Then x Θ , Φ y “ π i g ÿ a “ ¿ C a Θ Ξ r γ a s , (29) for all Θ P H p g q N .Proof. Let F “ α ´ p Ξ q be the Bers potential for Φ with cocycle Ξ. Then ij D Θ Φ R ´ N ω “ π i ij D d p Θ F q “ π i g ÿ a “ ¿ C a Θ Ξ r γ a s , much as in (28). (cid:3) Remark 2.2. ř ga “ ű C a Θ Ξ P r γ a s “ P . Corollary 2.1.
Let t Φ s u d N s “ be a H p g q N -basis and let t Φ _ r u d N r “ be the Petersson dual basis.For r, s “ , . . . , d N we have π i g ÿ a “ ¿ C a Φ _ r Ξ s r γ a s “ δ rs , (30) where Ξ s “ β p Φ s q is any cocycle representative associated with Φ s . Let t Φ s u d N s “ be a H p g q N -basis with potentials t F s u d N s “ and cocycles t Ξ s u d N s “ i.e. Ξ s is a particular cocycle representative associated with Φ s . Recall there exists a correspondingGEM form Ψ N of (22) determining the potentials t F s u d N s “ in (23). We then find We note that there appears to be a sign error in (4.1) of [McIT].
MICHAEL P. TUITE
Proposition 2.4. Ψ N and the cocycles t Ξ s u d N s “ obey π i g ÿ a “ ¿ C a Ψ N p¨ , y q Ξ s r γ a sp¨q “ Ξ s r λ sp y q , (31) where y P Ω p Γ q and λ is the unique Schottky group element such that λy P D .Proof. Ψ N p x, y q has a unique simple pole at x “ λy for x P D using (18). Let D ε “ D ´ ∆ ε where ∆ ε is a disc of arbitrarily small radius ε centered at λy with orientedboundary C ε . Parameterizing ∆ ε by x “ λy ` re i θ for 0 ď r ď ε and 0 ď θ ď π we findΨ N p x, y q ω “ p ρ p λy q ` O p ε qq dr ^ dθ for x P ∆ ε so that (23) implies the Bers potential F s “ α ´ p Ξ s q for Φ s is given by F s p y q “ ´ lim ε Ñ ij D ε Ψ N p¨ , y q Φ s R ´ N ω. Similarly to (16), we find that for fixed y π i d p Ψ N p¨ , y q F s q “ Ψ N p¨ , y q Φ s R ´ N ω, on D ´ t λy u Ą D ε . Then Stokes’ theorem implies ´ ij D ε Ψ N p¨ , y q Φ s R ´ N ω “ π i ÿ a P I ¿ C a Ψ N p¨ , y q F s ` π i ¿ C ε Ψ N p¨ , y q F s “ ´ π i g ÿ a “ ¿ C a Ψ N p¨ , y q Ξ s r γ a s ` π i ¿ C ε Ψ N p¨ , y q F s . Let x “ λy ` εe i θ on C ε so that Ψ N p x, y q F s p x q “ i p F s p λy q ` O p ε qq dθ implyinglim ε Ñ π i ¿ C ε Ψ N p¨ , y q F s “ F s p λy q . Combining these identities we find12 π i g ÿ a “ ¿ C a Ψ N p¨ , y q Ξ s r γ a s “ F s p λy q ´ F s p y q “ Ξ s r λ sp y q . (cid:3) Remark 2.3.
Since Ξ s r id s “
0, Proposition 2.4 implies that for y P D we have g ÿ a “ ¿ C a Ψ N p¨ , y q Ξ s r γ a s “ . (32)Conversely, (32) implies (31) using (24), (25) and (30).2.5. A canonically normalized basis for H p g q N . We define a canonical Z p Γ , H p q ´ N q homology basis t Ξ ak u for a P I ` and k “ , . . . , N ´ t γ b u for b P I ` , as followsΞ ak r γ b sp z q : “ δ ab z ka dz ´ N , (33) EROMORPHIC EXTENSIONS OF GREEN’S FUNCTIONS ON A RIEMANN SURFACE 9 with z a “ z ´ w a for Schottky parameter w a . For any given P P H p q ´ N with coboundaryΞ P we may write Ξ P r γ a sp z q “ N ´ ÿ k “ p ak z ka dz ´ N , a P I ` , for some complex coefficients p ak determined by P . Hence it follows thatΞ P “ g ÿ a “ N ´ ÿ k “ p ak Ξ ak , (34)since they coincide on each generator γ a . Thus there are 2 N ´ t Ξ ak u (cf. Lemma 2.2). Let Φ ak “ p ǫ ˝ α ´ q p Ξ ak q P H p g q N . Since p ǫ ˝ α ´ q p Ξ P q “ N ´ t Φ ak u given by g ÿ a “ N ´ ÿ k “ p ak Φ ak “ . (35)Let t Φ ak u J denote any H p g q N -basis indexed by p a, k q P J where J is a set, of cardinality d N , of distinct p a, k q values with a “ , . . . , g and k “ , . . . , N ´ t Ξ ak u J are independent modulo coboundaries. We let t Φ _ ak u J denote the Peterssondual basis and let Ψ Can N be the corresponding GEM form (22). Then (25), Corollary 2.1and Remark 2.3 imply Proposition 2.5. A H p g q N -basis t Φ ak u J with canonical cocycles t Ξ ak u J , Petersson dualbasis t Φ _ bl u J and GEM form Ψ Can N obeys ¿ C a Ψ Can N p x, y q Ξ ak r γ a sp x q “ , (36) Ψ Can N p x, y q ´ Ψ Can N p x, γ a y q “ ÿ p a,k qP J Φ _ ak p x q Ξ ak r γ a sp y q , (37) 12 π i ¿ C a Φ _ bl p x q Ξ ak r γ a sp x q “ δ ab δ kl , (38) for all y P D and p a, k q , p b, l q P J . Remark 2.4.
Proposition 2.5 is a natural generalization of the properties of the clas-sical differential of the third kind ω y ´ p x q : “ ÿ γ P Γ ˆ γx ´ y ´ γx ˙ d p γx q , which is a 1-differential is x and a 0-differential in y where ω y ´ p x q ´ ω γ a y ´ p x q “ ν a p x q for holomorphic 1-differential ν a normalized by ű C a ν b “ π i δ ab . However, Ψ N p x, y q hasa unique simple pole at x “ y whereas ω y ´ p x q has an additional simple pole at x “ N “ Variation of the Riemann Surface Moduli
We consider Bers potentials, holomorphic differentials and GEM functions for N “ § metric with line element ds „ | dz ` µ p z, z q dz | , for local coordinates z, z where | µ | ă B p z, z q : “ µ p z, z q dz ´ dz P A ´ , e.g.[GL]. B p z, z q is called a Beltrami differential. The metric can be transformed locallyto coordinates w, w where ds „ | dw | provided w p z, z q satisfies the Beltrami equation B z w “ µ B z w. (39)The mapping z Ñ w p z q is called a quasiconformal map. Define φ “ µρ , for the Poincar´emetric ρ , so that Φ p z q “ φ p z q dz P A , . B p z, z q is called an harmonic Beltramidifferential when B z φ “ p z q P H p g q , the space of holomorphic quadraticdifferentials. There is a 1-1 map between the infinitesimal variations of M g and thespace of harmonic Beltrami differentials i.e. a bijective antilinear map between themoduli tangent space T p M g q and H p g q e.g. [A].We may explicitly realise these ideas in the Schottky uniformization as follows. Con-sider a small variation in a Schottky parameter m Ñ m ` ε m with correspondingquasiconformal map given by z Ñ w p z, z, ε m q where w “ z ` ε m π f m ` O p ε m q , (40)for some f m p z, z q . (39) implies that µ “ ε m µ m ` O p ε m q where µ m “ π B z f m . Thus for an harmonic Beltrami differential F m “ f m p z q dz ´ P F is a Bers potentialfor Φ m “ µ m ρdz P H p g q from (14) with N “
2. The deformed Riemann surface isuniformized with a Schottky group Γ ε m where for each γ P Γ we define γ ε m P Γ ε m viathe compatibility condition: γ ε m w p z q “ w p γz q . But γ ε m z “ γz ` ε m B m p γz q ` O p ε m q and using (40) we find [EO, Ro, P] B m p γz q “ π p f m p γz q ´ f m p z qp γz q q “ π Ξ m r γ sp z q d p γz q , (41)where Ξ m denotes the 1-cocycle for potential F m . Define the following T p C g q basis: B a, : “ B w a , B a, : “ ρ a B ρ a , B a, : “ ρ a B w ´ a , (42)for a P I ` and ℓ “ , ,
2. For each generator γ b P Γ of (4) we have B a,k p γ b z q “ ρ a p z ´ w a q k ´ δ ab “ ´ Ξ ak r γ b sp z q d p γ b z q , where Ξ aℓ is the canonical cocycle basis of (33) for N “
2. Thus we find that B a,ℓ p γz q “ ´ Ξ aℓ r γ sp z q d p γz q , (43)for all γ P Γ. We have therefore established a natural pairing of B a,ℓ with Ξ aℓ . Inconjunction with Proposition 2.2 this implies Proposition 3.1. F » Z p Γ , H p q´ q » T p C g q as vector spaces. By Lemma 2.1 the coboundary space B p Γ , H p q´ q » H p q´ . H p q´ is trivially isomorphicas a vector space to the M¨obius sl p C q Lie algebra where P “ p p z q dz ´ P H p q´ for p P Π The factor of π is introduced to comply with our Bers potential definition (14). EROMORPHIC EXTENSIONS OF GREEN’S FUNCTIONS ON A RIEMANN SURFACE 11 is identified with ´ p p z qB z . By (43), the B p Γ , H p q´ q element Ξ P “ ř ℓ “ ř a P I ` p aℓ Ξ aℓ of(34) is paired with L C g P P T p C g q given by L C g P : “ ´ ÿ ℓ “ ÿ a P I ` p aℓ B a,ℓ “ ´ ÿ a P I p p W a qB W a , (44)for the original Schottky parameters W ˘ a . Thus t L C g P u generates the sl p C q subalgebraof T p C g q associated with the M¨obius action (8). In summary, we have the followingvector space isomorphisms: H p q´ » B p Γ , H p q´ q » sl p C q Ă T p C g q . (45)Recalling that S g “ C g { SL p C q with tangent space T p S g q “ T p M g q we may considerthe relevant quotients using Proposition 3.1 and (45) to find Proposition 3.2. H p g q » H p Γ , H p q´ q » T p M g q as vector spaces. We now discuss the geometrical meaning of a given N “ p x, y q .Following (24) we define Θ a p x ; ℓ q P H p g q for a P I ` and ℓ “ , , p x, y q ´ Ψ p x, γ a y q “ ÿ ℓ “ Θ a p x, ℓ qp y ´ w a q ℓ dy ´ , for each Γ generator γ a . It follows that for all γ P ΓΨ p x, γy q ´ Ψ p x, y q “ ´ ÿ ℓ “ ÿ a P I ` Θ a p x, ℓ q Ξ aℓ r γ sp y q , (46)for all γ P Γ. Hence t Θ a p x ; ℓ qu is a H p g q spanning set by (25). We also define a canonicaldifferential operator given by [GT1, TW] ∇ C g p x q : “ ÿ ℓ “ ÿ a P I ` Θ a p x, ℓ qB a,ℓ . (47) ∇ C g p x q is a holomorphic vector field on C g with quadratic differential coefficients.We next examine the dependence of ∇ C g p x q on the choice of GEM function Ψ . From(22) it is sufficient to consider a new GEM function q Ψ p x, y q “ Ψ p x, y q ´ Φ p x q P p y q for some Φ P H p g q and P P H p q´ . (46) implies q Θ a p x, ℓ q “ Θ a p x, ℓ q ` p aℓ Φ p x q forΞ P “ ř ℓ “ ř a P I ` p aℓ Ξ aℓ so that q ∇ C g p x q “ ∇ C g p x q ` Φ p x q L C g P . Therefore modulo sl p C q we find that ∇ C g p x q determines a unique vector field ∇ M g p x q independent of the choice of GEM function. Thus we may choose a basis of 3 g ´ tB aℓ u J P T p M g q , where J is a set of distinct p a, ℓ q values as describedin § t Ξ aℓ u J , H p q´ basis t Φ aℓ u J and Peterssondual basis t Φ _ aℓ u J . Then (37) implies that ∇ M g p x q “ ÿ p a,ℓ qP J Φ _ aℓ p x qB a,ℓ . (48) Furthermore, for any coordinates t m r u g ´ r “ on moduli space M g with T p M g q basis tB m r u g ´ r “ and corresponding H p g q -basis t Φ r u g ´ r “ with dual basis t Φ _ r u g ´ r “ then [O] ∇ M g p x q “ g ´ ÿ r “ Φ _ r p x qB m r . (49)Next let S g,n denote a Riemann surface with n punctures y , . . . , y n P S g with pa-rameter space C g,n : “ C g ˆ p S g q n . Define the differential operator [GT1, TW, O] ∇ C g,n p x q : “ ∇ C g p x q ` n ÿ k “ Ψ p x, y k q dy k B y k . (50)We note that (43) and (46) imply that for all γ P Γ ∇ C g p x qp γy q “ p Ψ p x, γy q ´ Ψ p x, y qq d p γy q , which implies that for each puncture y k ∇ C g,n p x qp γy k q “ Ψ p x, γy k q d p γy k q . (51) Proposition 3.3. ∇ C g,n p x q is a holomorphic vector field on C g,n for x P S g,n .Proof. We first show that ∇ C g,n p x q is a vector field on C g,n i.e. ∇ C g,n p x q is invariant under y k Ñ γy k for γ P Γ. Let p r w ˘ a , r ρ a , r y k q “ p w ˘ a , ρ a , γy k q . Then B a,ℓ “ r B a,ℓ ` n ÿ k “ B a,ℓ p r y k qB r y k , B y k “ B y k p r y k qB r y k . From (50) and (51) we obtain ∇ C g,n p x q “ r ∇ C g p x q ` n ÿ k “ ∇ C g,n p x qp r y k qB r y k “ r ∇ C g p x q ` n ÿ k “ Ψ p x, r y k q d r y k “ r ∇ C g,n p x q . Thus ∇ C g,n p x q is a vector field on C g,n . Furthermore ∇ C g,n p x q is holomorphic sinceΨ p x, y q and Θ a p x, ℓ q are holomorphic for x P S g,n . (cid:3) For a new GEM function q Ψ p x, y q “ Ψ p x, y q ´ Φ p x q P p y q we find that q ∇ C g,n p x q “ ∇ C g,n p x q ` Φ p x q L C g,n P , where by (44) L C g,n P : “ L C g P ´ n ÿ k “ p p y k qB y k “ ´ ÿ a P I p p W a qB W a ´ n ÿ k “ p p y k qB y k , i.e. L C g,n P generates an sl p C q subalgebra of T p C g,n q . Thus modulo sl p C q we find that L C g,n P determines a unique vector field ∇ M g,n p x q independent of the choice of GEMfunction. Similarly to the unpunctured case, we may choose a canonical cohomologybasis t Φ aℓ u J with corresponding canonical GEM form Ψ Can2 of Proposition 2.5 so that
Proposition 3.4.
There exists a unique holomorphic vector field on S g,n ∇ M g,n p x q “ ∇ M g p x q ` n ÿ k “ Ψ Can2 p x, y k q dy k B y k . EROMORPHIC EXTENSIONS OF GREEN’S FUNCTIONS ON A RIEMANN SURFACE 13
Remark 3.1.
The operators ∇ C g and ∇ C g,n appear in the conformal Ward identities fora vertex operator algebra on a Riemann surface as described in ref. [GT1] (for genus2) and in ref. [TW] (for genus g ). This leads to partial differential equations giving thevariation with respect to moduli of Riemann surface structures such as the bidifferentialof the second kind ω p x, y q , the projective connection, the holomorphic 1-differentials ν a and the period matrix Ω ab . Thus Rauch’s formula [Ra] is rederived and expressed as2 π i ∇ M g Ω ab “ ν a p x q ν b p x q , a, b P I ` . (52)In ref. [O] the operator ∇ M g p x q is defined and the operator ∇ M g,n p x q is anticipated butthe existence and a construction of the Ψ Can2 p x, y q term is not given. Similar partialdifferential equations for Riemann surface structures (including the prime form) arealso discussed in ref. [O].4. The Green’s Function G N p x, y q We consider the Green’s function G N p x, y q for the anti-holomorphic part of thePoincar´e metric compatible connection [Ma, McI, McIT] and its relationship to a GEMform Ψ N . We develop some novel properties for the Green’s function and describe anexplicit formula for the inverse Bers map α ´ employing the Green’s function.There is a unique connection acting on H p g q N compatible with the Poincar´e metric andthe complex structure defined by e.g. [McIT] B N ‘ B N : H p g q N Ñ H p g q N ` ‘ H p g q N, , where for Θ p z q “ θ p z q dz N P H p g q N we have B N θ p z q “ ρ p z q N B z ` ρ p z q ´ N θ p z q ˘ , B N θ p z q “ B z θ p z q . Note that ker B N “ H p g q N . Define the projection P N : H p g q N Ñ H p g q N by P N Θ : “ d N ÿ r “ x Θ , Φ r y Φ _ r , for Θ “ θ p z q dz N P H p g q N and any H p g q N -basis t Φ r “ φ r p z q dz N u d N r “ with Petersson dualbasis t Φ _ r “ φ _ r p z q dz N u d N r “ . It is also useful to define the projection kernel p N p x, y q : “ d N ÿ r “ φ _ r p x q φ r p y q ρ ´ N p y q , (53)so that P N θ p x q “ ť D p N p x, y q θ p y q d y .For N ě g ě
2, we define the Green’s function for B N to be a bidifferential p N, ´ N q -form G N p x, y q “ g N p x, y q dx N dy ´ N where the regular part defined by g RN p x, y q : “ g N p x, y q ´ x ´ y , (54)satisfies the following two conditions:(I) g RN p x, y q is holomorphic in x ,(II) g RN p x, y q is not meromorphic in y with 1 π B y g RN p x, y q “ p N p x, y q . Remark 4.1. (i) We may heuristically rewrite (II) as1 π B y g N p x, y q “ ´ δ p y ´ x q ` p N p x, y q , for Dirac delta function πδ p y ´ x q “ B y p y ´ x q ´ from (21). This is a definingproperty for the Green’s function for B N in the physics literature e.g. [EO, Ma].(ii) We may also write (II) in a coordinate-free way (similarly to (16)) where12 π i d p G N p x, ¨q Φ q “ Φ P N p x, ¨q , (55) on D ´ t x u for all Φ P H p g q N with P N p x, y q “ d N ÿ r “ Φ _ r p x q Φ r p y q R p y q ´ N ω p y q . Lemma 4.1.
The Green’s function is unique.Proof.
Suppose G N and r G N are Green’s functions for B N . Let G N p x, y q ´ r G N p x, y q “ h N p x, y q dx N dy ´ N , for h N p x, y q “ g RN p x, y q ´ r g RN p x, y q . (II) implies h N p x, y q dy ´ N is a holomorphic form ofweight 1 ´ N ă y (for fixed x ). Hence h N “ (cid:3) Proposition 4.1. x G N p¨ , y q , Φ y “ for all Φ P H p g q N .Proof. For a given H p g q N -basis t Φ r u d N r “ , let λ s p y q dy ´ N “ x G N p¨ , y q , Φ s y . Equation (21)implies that for x, y P D π B y λ s p y q “ π B y ij D g N p x, y q φ s p x q ρ p x q ´ N d x “ ´ φ s p y q ρ p y q ´ N ` π ij D B y g RN p x, y q φ s p x q ρ p x q ´ N d x. Then (53) and condition (II) imply1 π B y λ s p y q “ ´ φ s p y q ρ p y q ´ N ` d N ÿ r “ x Φ _ r , Φ s y φ r p y q ρ p y q ´ N “ . Thus λ s p y q dy ´ N is a negative weight holomorphic form and hence λ s “ (cid:3) We now construct the unique Green’s function [Ma, McIT]. Let t Φ s u d N s “ be a H p g q N -basis with potentials t F s p y qu d N s “ , Petersson dual basis t Φ _ r u d N r “ and GEM form Ψ N . Proposition 4.2.
The Green’s function G N p x, y q for x, y P D is given by G N p x, y q “ Ψ N p x, y q ` d N ÿ r “ Φ _ r p x q F r p y q . (56) Proof. Ψ N p x, y q ` ř r Φ _ r p x q F r p y q is an N -differential with respect to x and using (17)and (25), it is a 1 ´ N differential with respect to y . Thus condition I of (54) is verified.It is straightforward to confirm condition II using the Bers equation (14) and that ψ RN p x, y q “ ψ N p x, y q ´ x ´ y , is holomorphic in y . (cid:3) EROMORPHIC EXTENSIONS OF GREEN’S FUNCTIONS ON A RIEMANN SURFACE 15
Remark 4.2.
We note that Proposition 4.1 is easily verified using (23). Furthermore,we can interpret the Green’s function as being the orthogonal projection of the GEMform Ψ N p x, y q . Conversely, Ψ N p x, y q can be interpreted as a meromorphic extension ofthe unique Green’s function G N p x, y q which is why we refer to Ψ N p x, y q as a Green’sfunction with Extended Meromorphicity or GEM form.The inverse map α ´ (associated with the inverse Bers map β ´ ) which exists byProposition (2.2) can be explicitly described in terms of the Green’s function as follows: Proposition 4.3.
Let Ξ be a 1-cocycle. Then Φ “ β ´ p Ξ q has Bers potential F Ξ p y q “ α ´ p Ξ q given by F Ξ p y q “ π i g ÿ a “ ¿ C a G N p¨ , y q Ξ r γ a s ´ Ξ r λ sp y q , (57) where λ P Γ is the unique Schottky group element such that λy P D .Proof. Let t Φ s u d N s “ be a H p g q N -basis with potentials t F s p y qu and corresponding cocyclebasis t Ξ s u , Petersson dual basis t Φ _ r u and GEM form Ψ N . Write Ξ “ ř d N r “ x r Ξ r where x r “ π i g ÿ a “ ¿ C a Φ _ r Ξ r γ a s , from Corollary 2.1. Then (56) and Proposition 2.4 imply12 π i g ÿ a “ ¿ C a G N p¨ , y q Ξ r γ a s “ π i g ÿ a “ ¿ C a Ψ N p¨ , y q Ξ r γ a s ` d N ÿ r “ x r F r p y q“ Ξ r λ sp y q ` F Ξ p y q , where F Ξ p y q “ ř d N r “ x r F r p y q . (cid:3) Remark 4.3. (i) Define F D p y q “ π i ř ga “ ű C a G N p¨ , y q Ξ r γ a s for y P Ω p Γ q . It follows that F Ξ p y q “ F D p y q for all y P D since Ξ r id s “ F D | γ “ F D for all γ P Γ since G N p¨ , y q is a 1 ´ N form in y . Furthermore,(57) implies that for all γ P Γ and y P Ω p Γ q F Ξ p γy q “ F D p y q ´ Ξ r λγ ´ sp γy q . since λγ ´ p γy q P D . Thus F Ξ | γ ´ F Ξ “ ´ Ξ r λγ ´ s| γ ` Ξ r λ s “ Ξ r γ s using the1-cocycle condition (12). Lastly, (57) implies that F Ξ satisfies (16) using (55).(iii) Proposition 4.3 can be adapted to the case of any Kleinian group with contourintegrals over appropriate boundaries of the disconnected components of the fun-damental domain D as in Remark 2.1.(iv) (57) provides the motivation for a further generalization of the notion of a GEMform associated with a non-Kleinian uniformization of a genus two Riemann sur-face formed by sewing two tori [GT2]. References [A] Ahlfors, L. Some remarks on Teichm¨uller’s space of Riemann surfaces, Ann.Math. (1961)171-191.[Be1] Bers, L. Inequalities for finitely generated Kleinian groups, J.Anal.Math. (1967) 23-41.[Be2] Bers, L. Eichler integrals with singularities, Acta.Math. (1971) 11–22. [Be3] Bers, L. Automorphic forms for Schottky groups, Adv.Math. (1975) 332-361.[Bo] Bobenko, A. Introduction to compact Riemann surfaces, in Computational Approach to Rie-mann Surfaces , edited Bobenko, A. and Klein, C., Springer-Verlag, Berlin-Heidelberg (2011).[EO] Eguchi, T. and Ooguri, H. Conformal and current algebras on a general Riemann surface,Nucl.Phys.
B282
Riemann Surfaces . Springer-Verlag, Berlin-New York (1973).[Fa] Fay, J.
Theta Functions on Riemann Surfaces . Lect. Notes Math. , Springer-Verlag, Berlin-New York (1973).[Fo] Ford, L.R.
Automorphic Functions . AMS-Chelsea, Providence (2004).[G] Gardiner, F.P. Automorphic forms and Eichler cohomology, in
A Crash Course on KleinianGroups edited by Bers, L. and Kra, I., Lect. Notes Math.
Springer, Berlin (1974).[GL] Gardiner, F.P. and Lakic, N.
Quasiconformal Teichm¨uller Theory , Mathematical Surveys andMonographs , AMS, Providence (1991).[GR] Gunning, R.C and Rossi, H. Analytic Functions of Several Complex Variables . AMS-Chelsea,Providence (2009).[GT1] Gilroy, T. and Tuite, M.P. Genus two Zhu theory for vertex operator algebras,arXiv:1511.07664. Under revision.[GT2] Gilroy, T. and Tuite M.P. A meromorphic extension of Green’s functions on a genus twoRiemann surface formed from sewn tori. To appear.[Ma] Martinec, E. Conformal field theory of a (super-)Riemann surface, Nucl.Phys.
B281 (1987)157–210.[McI] McIntyre, A. Analytic torsion and Faddeev-Popov ghosts, SUNY PhD thesis 2002,hdl.handle.net/11209/10688.[McIT] McIntyre, A. and Takhtajan, L.A. Holomorphic factorization of determinants of Laplacians onRiemann surfaces and a higher genus generalization of Kroneckers first limit formula, GAFA,Geom. Funct. Anal. (2006) 1291–1323.[Mu] Mumford, D. Tata Lectures on Theta I and II . Birkh¨auser, Boston (1983).[O] Odesskii, A. Deformations of complex structures on Riemann surfaces and integrable structuresof Whitham type hierarchies, arXiv:1505.07779.[P] Playle, S. Deforming super Riemann surfaces with gravitinos and super Schottky groups,J.H.E.P. (2016) 035.[Ra] Rauch, H.E. On the transcendental moduli of algebraic Riemann surfaces, Proc.Nat.Acad.Sc. (1955) 42–48.[Ro] Roland, K. Beltrami differentials and ghost correlators in the Schottky parametrization,Phys.Lett. B312 (1993) 441-450.[TW] Tuite, M.P. and Welby, M. General genus Zhu recursion for vertex operator algebras,arXiv:1911.06596.
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