Two remarks on Riemann surfaces

Publicacions Matemàtiques, Vol 38 (1994), 463-477 . TWO REMARKS ON RIEMANN SURFACE S J . M . RoDRÍGUEZ * Abstxact  We study the relationship between linear isoperimetric inequalitie s and the existence of nonconstant positive harmonic functions o n Riemann surfaces . We also study the relationship between growth conditions o f length of spheres and the existence of Green's function on Rieman n surfaces . 1 . Introductio n The theme of this paper is to clarify some connections between differ - ent conformal invariants of Riemann surfaces . By a Riemann surface R we denote a twodimensional surface wit h a complete metric of constant negative curvature -1 . In this case, th e universal covering space of R is the unit disk q ; R is endowed wit h its Poincaré metric, i .e ., the metric obtained by projecting the Poincar é metric of the unit disk ds = 2(1 - -- I z 2 ) 1 ldz I onto R . The unit disk q is isornetric to the upper-half plane U endowed with its Poincaré metri c ds = idz l/y, z = x + i y E U : We will use both models of the hyperboli c plane . The only Riemann surfaces which are left out are the sphere, th e plane, the punctured plane and the tori . We shall say that a Riemann surface R satisfies a linear isoperimetri c inequality ( LII ) if there exists a finite constant h(R) so that for ever y bounded open set G with smooth boundary we hav e A(G) ç h(R)L(5G) . Here and from now on, A, L and d refer to Poincaré area, length an d distance of R . There are close connections between LII and some conformal invariant s on Riemann surfaces, namely the bottom of the spectrum of the Laplace - Beltrami operator b(R), and the exponent of convergence 6(R) . Thes e connections are described in the next two known results . *Research supported in part by a grant of the DGICYT, Ministerio de Educación y Ciencia, Spain . 464  J . M . RODRÍGUE Z Theorem A [Ch], [Bu, p . 228], [FR1] . A Riemann surface R satis - fies a linear isoperimetric inequality if and only if b(R)> O . In fact , 4 < b(R)h(R) 2 and b(R)h(R) < 2 The next result is a well known theorem of Elstrodt-Patterson - Sulli - van : Theorem B [S, p . 333] . A Riemann surface R satisfies a linea r isoperimetric inequality if and only if 6(R) < 1 . In fact , b(R)  1 1 6(R)(1 -6(R)}, if 2 Ç á(R ) Ç 1 . We shall be particularly interested in the class 13 of Riemann surface s which do not satisfy a linear isoperimetric inequality . A theorem of Myrberg [T, p . 522] states that if 6(R) < 1 (if R satisfies a LII ) then R has a Green's function (R DG in the language o f classification theory) . If R is a plane domain (in fact ; if R is a surface o f almost finite genus [SN, p . 1931), the following conditions are equivalen t [SN, p . 1941 : (i) R has Green ' s function (R aG ) , (u) R has a n o n - constant positive harmonic function (R O HP) , (iii) R has a non -cosntant bounded harmonic function (R O HB) , (iv) R has a nonconstant harmonic function with finite Dirichlet in - tegral (R OHD ) . The Dirichlet integral of a function is the square of the L 2 -norm of it s gradient . For arbitrary Riemann surfaces the following strict inclusions are wel l known : VG C V HP C OHB Ca` HD . One would like to understand exactly how the class 13 fits into thi s chain . As we have said aboye, in the case of surfaces of almost finite genus , OG = O HP = O HB = = a HD C 13 . The inclusion is strict, even in th e planar case, as it is shown by the example Ro = q \(U 1 {2_ k } U {O} ) Ro « 0 HD because it is a plane domain whose boundary has positiv e logarithmic capacity [T, p . 81] ; Ro E I3 because U 1 {2} U {O} is a discrete set with an accumulation point in q [FR1, Theorem 4j . Thi s example shows that L3 is not contained in OHP, OHB or 0H D . 4,  if 0 < 6(R) < ~, Two REMARKS ON RIEMANN SURFACES  46 5 The inclusion UH DC U is nat true in general . To describe why , consider thereal linear space HD(R) of harmonic functions in R wit h finite Dirichlet integral . Then, we have the followin g Theorem C [Ro] . Let R be a Riemann surface which satisfies a linea r isoperimetric inequality. If there exists in R a set of disjoint simple close d curves {Yj}i, such that R\ U j ry j contains n connected components o f infinite area, then dim HD(R) ~ n . This inequality is best possible : for each n> 1, there exists a Riemann surface R n satisfying the hypothesis of the theorem and such tha t dim HD(R n ) = n . In particular, for n = 1 we have the followin g Corollary . There is a Riemann surface R 1 with .LH and such tha t every harmonic function with finite Dirichlet integral on R 1 is constant . R 1 can be constructed verifying the extra customary hypothesis o f bounded geometry [K], which in our context simply means that th e injectivity radius t(R1) is positive . For any Riemann surface R, t(R) i s defined as t(R) = inf{t(p) : p E R} , where t(p) is the injectivity radius of the geodesic exponential map centered at p . Summarizing, the situation is the following : OG C 8, QG C0H p C 0H B C 0H D , and there are no inclusion relationships between Z3 and 0 HD . But, i s there any inclusion relationship between L3 and a H$ ar between L3 and 0HP ? It was natural to expect the inclusion D HB C IJ to be true . Fo r instance, if R is a surface of almost finite genus, then Q HB = OG C Z3 ; if R has a LII and is a regular covering of a compact surface (the n R has infinite genus), one can prove that the Varopoulos's index x(R ) [V3] is positive, and this implies the existence of nonconstant bounde d harmonic functions . Then, the inclusion is true for finite genus and fo r two extremal cases with infinite genus : almost finite genus and coverin g spaces of compact surfaces . In this paper we prove by an example that the inclusion 0HB C !J i s not true . In fact, we prove that the weaker inclusion O H p c Z3 is no t true, even with the hypothesis t ~ O : 406  J . M . RODRÍGUE Z Theorem 1 . There exists a Riemann surface M with LII, t(M) > O , and such that every positive harmonic function on M is constant . A most interesting example in the classification of Riemann surface s was obtained by Lyons [L] . He constructed two quasiisometric (and, i n particular, quasiconformaly equivalent) Riemann surfaces, one of the m belonging to a H p while the other supports nonconstant bounded harmonic functions . We shall be using his construction as a basic ingredien t in the proof of theorem 1 . The relationship between growth conditions of area of balls and th e existence of Green ' s function is a well-known and classical issue . W e refer, far instance, to [D], [E], [F], [FR2], [G], [Ka], [LS], [Vl] and th e references therein far some general (in arbitrary Riemannian manifolds ) geometric and topological conditions related to the existence of Gree n ' s function . Green's function exists if and only if there exists a positive non - constant superharmonic function or equivalently if Brownian motion o n the surface is transient (see, e .g . [AS, p . 204], [V1]} . If p is a point of a Riemann surfaceR and t is a positive number w e denote by A R (p, t) and L R (p, t), the area of B(p, t) and the length of a B (p, t) respectively, where B (p, t) is the ball of radius t centered at p . Of course, A R (p,t) Ç A q (o, t) = 47r sinh 2 (t/2) 7re t as t--4 oo, an d L R (p, t) Ç L q (0, t) = 27r sinh tP .,- 7re t as t - --} oo . The following theorem is known . Theorem D . (i) If for a point po E R and constants co, t o A R (po a t) ? coe t , f or every t ~ to , then R has a Green's function . Given a function zf~ : (o, oc) —> (0, oo) , increasing, and such tha t lim 1P(t ) t—oo e t there exists a Riernann surface R and a point po E R so that AR(po, t) ~ O(t) , for every t~t i , but R has no Green's function . Part (i) is elementary and part (u) is due to Nicholls [N] . We do remar k that in the example of (u) of theorem D, Rcan be chosen to be planar . Two REMARKS ON RIEMANN SURFACES  46 7 It has been suggested by N . Varopoulos that in our situation, i .e . , constant negative curvature, a u nif orm exponential growth o f the cre a should imply that Green ' s function exists . More precisely, if there ar e positive constants c, a, to, such tha t A R (p, t) ~ ce' t , for every t ~ to, p E R , can we deduce that R has a Green's function ? The following theorem [FR2] answers the question . Theorem E . (i) If R is a noncompact Riemann surface of finit e genus and there are positive constants c, to, such that A R (p, to ) ~ c, f or every p E R , then R has a Green's function . (u) There exists a Riemann surface R so that A R (p,t) ~ c e ° t , for every t to, p E R , where c, rxo, to are positive numbers, and such that R does not have a Green's function . Varopoulos in [V2, p . 271] gives an example like the one in theorem E with a o = 1/2, but with variable curvatura . Notice that topologicall y his example is the plane . The example constructed in [FR2] has exponential growth of area o f bálls but the length of spheres does not grow . As a matter of fact, ther e exists a point p and positive numbers t n converging to infinity, suc h that L (p, t n ) remain bounded as n —> oo . And actually that is why th e Riemann surface has no Green's function . Therefore, it is natural to ask if the stronger conditio n L R (p, t) ~ ce at , for every t ~ to, p E R , implies that R has a Green's functión . Observe that this implie s A R (p, t) ~ c,e at , for every t ~ to + e, p E R, e > o , because A R (p, t) = j o L R (p, s) ds . Observe that if L R (po, t) ~ ce t , for some point po E R and for ever y t ~ to then, of course, R has a Green's function, while, on the othe r hand, a classical criterion of Ahlfors {A] says that if R has a Green' s function then L R(po, t), grows rapidly ; in fact , dt C oo . L R ( p o ) t } Here we prove 468  J . M . RODRÍGUE Z Theorem 2 . There exists a Riemann surface N so that L N (p, t) ~ ce t/ 2 , for every t ~ to, p E Ñ , where c, to are positive numbers, and such that,N does not have a Green' s function . Notice in particular that theorem 2 has theorem E, (u) as an immediat e corollary . The proof of theorem 2 is simpler and gives the precise rat e ao = 1/2 . We do not know what the sharp rate ao in theorem E is . The organization of the paper is as follows . In Section 2 we prove the - orem 1 . In Section 3 we construct the Riemann surfacé .IIN of theorem 2 , while in Section 4 we prove that if satisfies the inequality LN(p, t) ~ ce t/2 . In Section 5 we prove that N has no Green's function . 2 . Proof of Theorem 1 The desired Riemann surface M will be obtained with the help of two graphs G and Go . In the set of vertices of any connected graph we can define a natura l distance : d(p, q) = inf{length of the paths from p to q} , where all edges have length 1 . This will be "the distance " in all graph s of this section . The degree of a vertex is the number of its neighbours , i .e ., the number of vertices at distance 1 from it . The vertices of the graphs G and Go are simply the elements of th e free group on two generators I` = (a, b) . Now we define the edges of G and Go . Each element of I can be expressed as a word with the letters a, a —1 , b, b' . This expression is unique if no simplification of contiguous letter s is possible ; in this case we say that the word is a reduced word . For each n, we choose B 7z a length preserving permutation of the word s of length greater or equal than n . This permutation must have orde r 3n—1 and must satisfy that g and g8 n start on the left with the sam e letter for all reduced words g of length n . If g is a reduced word of lengt h at least n, there is a unique factorization g = g l g 2 with length ( g i) -} - length (92) = length (g), and length (92) = n . We also require that 8 n satisfies g8n = g 1 ( g 2 e n) (observe that length (g8n) = length (g), becaus e On does not change the leftmost element of g 2 ) . Two REMARKS ON RIEMANN SURFACES  46 9 The sequence {0 n }11 1 can be arranged so that it verifies the recursio n ( 0 n) 3 = 0 n— 1 g l = identity . Let {r} and {s} be two sequences of natural numbers satisfyin g n < rn < s n <  and s n — r n ~ ( 3 n—1 ) 2+ ', for some positive E and for each n . We define the set of edges of G in the foliowing way . If g 1 and g 2 ar e reduced words of I, there is an edge between them if and only if they verify one of the conditions : ( i) g 1 g 1 2 -- a ±1 , (u) g1g 2 1 = b + 1 (iii) g 1 = g 2 0, n ±1 and length (Yl) E [r n , s n ] for some n . Consequently, every vertex of G has degree four or six . Go is simply the Cayley graph of the group r, i .e ., g1 and g 2 ar e connected by an edge of Go only if (i) or (u) is satisfied . Then, every vertex of Go has degree four . A way to build up our Riemann surface M, modelled upon the grap h G, is to use the so called Ldbell Y -pieces, which are a standard too l for constructing Riemann surfaces . A clear description of these Y -piece s and their use is given in [C, Chapter X .31 . (In [L] there is an equivalen t way of constructing Riemann surfaces . ) A Ldbell Y -piece is a three-holed sphere, endowed with a metric of constant negative curvature -1, so that the boundary curves are geodesics . The Y -pieces are a flexible tool : given any three positive numbers a, b , c, there is a Y -piece with boundary curves which have lengths a, b, c (see [C, p . 248] and [Fe, p . 99] far details) . A X-piece (*-piece) is a four-holed (six-holed) sphere, endowed with a metric of curvature -1, so that the boundary curves are geodesics . W e can construct these pieces, for example, joining two (four) Y -pieces, b y identifying corresponding boundary curves of the same length, in such a way that the resulting surface has genus zero . We are free to choose the lengths of the four (six) boundary curves o f the X--piece (*-piece). We choose the lengths of the boundary curves o f a X -piece and a * -piece following the construction of the manifold M in [L, p . 57] (a large "cylinder" in M correspondsto a short boundar y curve of our Riemann surface M [R]) . The four boundary geodesics o f a X -piece have the same length a . Four of the boundary geodesics o f a * -piece also have length a ; the order two boundary curves have the 470  J . M . RODRÍGUE Z same length with ,C3 a . W e take now infinitely many copies of bot h pieces . W e now build M by joining these pieces foliowing the combinatoria l design of G, with the X -pieces (*-pieces) in the place of the vertices o f degree four (six), glueing boundary curves of the same length . M is a complete surface of constant negative curvature -1, and since we hav e used only two distinct pieces to build up M, it is obvious that t(M) > O . The results of [L] give that every positive harmonic function on M i s constant . To prove that M has a LII we need to make precise the metric relationship between G and M . Following Kanai ' s terminology [K], we sa y that a mapping not necessarily continuous, between two metric space s : ( M i, d i) - 4 . ( M 2 5 d a ) is a rough isometry if the following two conditions are satisfied : (i) There are constants a > 1 and b > 0 such tha t a —l di (x, y) — b < d 2 (( P ( x ), (P( y ))  ad i (x, y) + b , forallx,yEM 1 . (u) For some e > 0, the e -neighbourhood of (P( M 1 ) covers M 2 . A metric space M 1 is said to be roughly isometric to a metric spac e M 2 if there exists a rough isometry from IVI 1 into M 2 . It can be checke d that being roughly isometric is an equivalence relation between metri c spaces . It is obvious that the graph G and the surface M are roughly isometric . If F is a graph, and P is a subset of vertices of F we define its boundar y aP by aP {v E V(F) : d(v,P) = 1} . If ~ . 1 denotes the number of elements of a subset of vertices, the linea r isoperimetric constant of F is defined b y h(F) = sup 15 1 I P 1 1 ' where P ranges over all the non -empty finite subsets of vertices of F . Lemma K . Let R be a Riemannian manifold with bounded geometr y and let F be a graph with bounded degree . If R and F are roughly isometric, then R satis , fies a linear isoperimetric inequality if and only if F satisfies a linear isoperimetric inequality . Two REMARKS ON RIEMANN SURFACES  47 1 To prove this lemma it is enough to combine two lemmas of Kanai [K , p . 401] (observe that the hypothesis of bounded geometry is satisfied i n our case, and also that m = oo is allowed in the notation of Kanai) . Lernma K says that the surface M and the graph G simultaneousl y verify or not a LII . Moreover, the definition of the linear isoperimetri c constant in a graph implies that G has a LII if the tree Go has a LII , because both have the same vertices and G has more edges . It is elementary that a regular tree of degree d > 3, satisfies the LI I with constant 1/(d— 2) . In particular, Go satisfies the LI I (1) Lemma K then gives that our surface M satisfies LII . The proof o f theorem 1 is complete . ■ 3 . Constructing the Riemann surface N Let U be the upper-half plane U = {x + iy E C : y > o} endowe d with its Poincaré metric ds = Idz Í /y . Let Vo be the closed subset o f {x + iy : -1 Ç x Ç 1} limited by the geodesic ares g 1 , g2, g 3 : = {1 +iy : y ~ l b g 2 = {—1 + iy : y ~ 1} , g 3 ={x+iy :x 2 +y 2 =2,—1<x<1} . We define V k = T k V o, for all integer k, where T k (z) = z + 2k, an d let W 1 be the closed periodic set W 1 = UkVk . The boundary of W 1 i s the union of the geodesic ares G l = T k (g3 ) . Observe that two geodesic s G~ and G~ +1 meets at the point 2k + 1 + i with angle 7 r / 2 . We denot e yi = Gi k and 771 = G r+1 . Let W 2 be another copy of W 1 . We denote by G2 , '-y1,  the analogue s in W 2 of Gl , ry1, r~1, respectively . If we join both copies W 1 and W 2 identifying  with 771, we obtain a Riemann surface W with boundary . Let 7k be the simple closed geodesic s in W, ry k = y1 U yl ; then the boundary curves of W are { Let W' be another copy of W . We denote by y' k the analogues in W ' of y k . Finally, if we join W and W ' identifying 7k with ry' k , we obtai n the complete Riemann surface ,11Ï .