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Journal articles on the topic 'Hyperbolic Riemann surfaces'

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Published: 6 September 2023

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1

Aulaskari, Rauno, and Huaihui Chen. "On Classes for Hyperbolic Riemann Surfaces." Canadian Mathematical Bulletin 59, no. 01 (March 2016): 13–29. http://dx.doi.org/10.4153/cmb-2015-033-8.

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AbstractThe Qpspaces of holomorphic functions on the disk, hyperbolic Riemann surfaces or complex unit ball have been studied deeply. Meanwhile, there are a lot of papers devoted to theclasses of meromorphic functions on the disk or hyperbolic Riemann surfaces. In this paper, we prove the nesting property (inclusion relations) ofclasses on hyperbolic Riemann surfaces. The same property for Qp spaces was also established systematically and precisely in earlier work by the authors of this paper.

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2

Jorgenson, Jay, and Rolf Lundelius. "hyperbolic Riemann surfaces of finite volume." Duke Mathematical Journal 80, no. 3 (December 1995): 785–819. http://dx.doi.org/10.1215/s0012-7094-95-08027-2.

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3

Hu, Zhi, and Runhong Zong. "Hyperbolic Superspaces and Super-Riemann Surfaces." Communications in Mathematical Physics 378, no. 2 (July 16, 2020): 891–915. http://dx.doi.org/10.1007/s00220-020-03801-5.

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4

Ji, Lizhen. "Spectral degeneration of hyperbolic Riemann surfaces." Journal of Differential Geometry 38, no. 2 (1993): 263–313. http://dx.doi.org/10.4310/jdg/1214454296.

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5

Haas, Andrew. "Diophantine approximation on hyperbolic Riemann surfaces." Acta Mathematica 156 (1986): 33–82. http://dx.doi.org/10.1007/bf02399200.

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6

Kong, De-Xing, Kefeng Liu, and De-Liang Xu. "The Hyperbolic Geometric Flow on Riemann Surfaces." Communications in Partial Differential Equations 34, no. 6 (May 14, 2009): 553–80. http://dx.doi.org/10.1080/03605300902768933.

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7

Parlier, Hugo. "The homology systole of hyperbolic Riemann surfaces." Geometriae Dedicata 157, no. 1 (May 8, 2011): 331–38. http://dx.doi.org/10.1007/s10711-011-9613-0.

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8

Keen, Linda. "Hyperbolic Geometry and Spaces of Riemann Surfaces." Mathematical Intelligencer 16, no. 3 (June 1994): 11–19. http://dx.doi.org/10.1007/bf03024351.

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9

Yanagishita, Masahiro. "Completeness of 𝑝-Weil-Petersson distance." Conformal Geometry and Dynamics of the American Mathematical Society 26, no. 3 (May 10, 2022): 34–45. http://dx.doi.org/10.1090/ecgd/369.

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Our goal of this paper is to research the completeness of the p p -Weil-Petersson distance, which is induced by the p p -Weil-Petersson metric on the p p -integrable Teichmüller space of hyperbolic Riemann surfaces. As a result, we see that the metric is incomplete for all the hyperbolic Riemann surfaces with Lehner’s condition except for the ones that are conformally equivalent to either the unit disk or the punctured unit disk. The proof is based on the one by Wolpert’s original paper, which is given in the case of compact Riemann surfaces.

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10

Baik, Hyungryul, Farbod Shokrieh, and Chenxi Wu. "Limits of canonical forms on towers of Riemann surfaces." Journal für die reine und angewandte Mathematik (Crelles Journal) 2020, no. 764 (July 1, 2020): 287–304. http://dx.doi.org/10.1515/crelle-2019-0007.

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AbstractWe prove a generalized version of Kazhdan’s theorem for canonical forms on Riemann surfaces. In the classical version, one starts with an ascending sequence {\{S_{n}\rightarrow S\}} of finite Galois covers of a hyperbolic Riemann surface S, converging to the universal cover. The theorem states that the sequence of forms on S inherited from the canonical forms on {S_{n}}’s converges uniformly to (a multiple of) the hyperbolic form. We prove a generalized version of this theorem, where the universal cover is replaced with any infinite Galois cover. Along the way, we also prove a Gauss–Bonnet-type theorem in the context of arbitrary infinite Galois covers.

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11

Janardhanan, Jaikrishnan. "Proper holomorphic mappings between hyperbolic product manifolds." International Journal of Mathematics 25, no. 04 (April 2014): 1450039. http://dx.doi.org/10.1142/s0129167x14500396.

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We prove a result on the structure of finite proper holomorphic mappings between complex manifolds that are products of hyperbolic Riemann surfaces. While an important special case of our result follows from the ideas developed by Remmert and Stein, the proof of the full result relies on the interplay of the latter ideas and a finiteness theorem for Riemann surfaces.

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12

Divakaran, Divakaran, and Jaikrishnan Janardhanan. "Finiteness theorems for holomorphic mapping from products of hyperbolic Riemann surfaces." International Journal of Mathematics 28, no. 07 (May 29, 2017): 1750060. http://dx.doi.org/10.1142/s0129167x17500604.

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We prove that the space of dominant/non-constant holomorphic mappings from a product of hyperbolic Riemann surfaces of finite type into certain hyperbolic manifolds that can be covered by a bounded domain is a finite set.

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13

Portilla, Ana, José M. Rodríguez, and Eva Tourís. "THE ROLE OF FUNNELS AND PUNCTURES IN THE GROMOV HYPERBOLICITY OF RIEMANN SURFACES." Proceedings of the Edinburgh Mathematical Society 49, no. 2 (May 30, 2006): 399–425. http://dx.doi.org/10.1017/s0013091504001555.

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AbstractWe prove results on geodesic metric spaces which guarantee that some spaces are not hyperbolic in the Gromov sense. We use these theorems in order to study the hyperbolicity of Riemann surfaces. We obtain a criterion on the genus of a surface which implies non-hyperbolicity. We also include a characterization of the hyperbolicity of a Riemann surface $S^*$ obtained by deleting a closed set from one original surface $S$. In the particular case when the closed set is a union of continua and isolated points, the results clarify the role of punctures and funnels (and other more general ends) in the hyperbolicity of Riemann surfaces.

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14

BARDEN, DENNIS, and HYUNSUK KANG. "Isospectral surfaces of genus two and three." Mathematical Proceedings of the Cambridge Philosophical Society 153, no. 1 (February 27, 2012): 99–110. http://dx.doi.org/10.1017/s0305004112000126.

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AbstractWe give examples of isospectral non-isometric surfaces of genus two and three with variable curvature, as well as hyperbolic orbifolds of genus two. We apply the first result to construct isospectral potentials on a Riemann surface of genus two.

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15

Monastyrsky, Michael, and Sergei Nechaev. "Correlation Functions for Some Conformal Theories on Riemann Surfaces." Modern Physics Letters A 12, no. 09 (March 21, 1997): 589–96. http://dx.doi.org/10.1142/s0217732397000613.

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We discuss the geometrical connection between 2-D conformal field theories, random walks on hyperbolic Riemann surfaces and knot theory. For the wide class of CFTs with monodromies being the discrete subgroups of SL(2, ℝ), the determination of four-point correlation functions are related to construction of topological invariants for random walks on multipunctured Riemann surfaces.

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16

Fernandez, J. L., and M. V. Melian. "Bounded Geodesics of Riemann Surfaces and Hyperbolic Manifolds." Transactions of the American Mathematical Society 347, no. 9 (September 1995): 3533. http://dx.doi.org/10.2307/2155022.

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17

Schneider, Matthias. "Closed magnetic geodesics on closed hyperbolic Riemann surfaces." Proceedings of the London Mathematical Society 105, no. 2 (April 3, 2012): 424–46. http://dx.doi.org/10.1112/plms/pds013.

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18

Fernández, J. L., and M. V. Melián. "Bounded geodesics of Riemann surfaces and hyperbolic manifolds." Transactions of the American Mathematical Society 347, no. 9 (September 1, 1995): 3533–49. http://dx.doi.org/10.1090/s0002-9947-1995-1297524-7.

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19

Masumoto, Makoto. "Hyperbolic lengths and conformal embeddings of Riemann surfaces." Israel Journal of Mathematics 116, no. 1 (December 2000): 77–92. http://dx.doi.org/10.1007/bf02773213.

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20

Akrout, Hugo, and Bjoern Muetzel. "Construction of hyperbolic Riemann surfaces with large systoles." Journal of Geometry 107, no. 1 (August 20, 2015): 187–205. http://dx.doi.org/10.1007/s00022-015-0288-9.

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21

Aulaskari, Rauno, Yuzan He, Juha Ristioja, and Ruhan Zhao. "Qp Spaces on Riemann Surfaces." Canadian Journal of Mathematics 50, no. 3 (June 1, 1998): 449–64. http://dx.doi.org/10.4153/cjm-1998-024-4.

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AbstractWe study the function spaces Qp(R) defined on a Riemann surface R, which were earlier introduced in the unit disk of the complex plane. The nesting property Qp(R) ⊆Qq(R) for 0 < p < q < ∞ is shown in case of arbitrary hyperbolic Riemann surfaces. Further, it is proved that the classical Dirichlet space AD(R) ⊆ Qp(R) for any p, 0 < p < ∞, thus sharpening T. Metzger's well-known result AD(R) ⊆ BMOA(R). Also the first author's result AD(R) ⊆ VMOA(R) for a regular Riemann surface R is sharpened by showing that, in fact, AD(R) ⊆ Qp,0(R) for all p, 0 < p < ∞. The relationships between Qp(R) and various generalizations of the Bloch space on R are considered. Finally we show that Qp(R) is a Banach space for 0 < p < ∞.

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22

Ito, Manabu, and Hiroshi Yamamoto. "Holomorphic mappings between compact Riemann surfaces." Proceedings of the Edinburgh Mathematical Society 52, no. 1 (February 2009): 109–26. http://dx.doi.org/10.1017/s0013091507000223.

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AbstractThere are only finitely many non-constant holomorphic mappings between two fixed compact Riemann surfaces of genus greater than 1. This result goes under the name of the de Franchis theorem. Having seen that the set of such holomorphic mappings is finite, we naturally want to obtain a bound on its cardinality. It has been known for some time that there exist various bounds depending only on the genera of the surfaces. Here we obtain ‘better’ bounds of the above type, using arguments based on the rigidity of holomorphic mappings and the hyperbolic geometry of surfaces.

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23

CRUZ-COTA, ALDO-HILARIO, and TERESITA RAMIREZ-ROSAS. "THE COMPLEXITY OF RIEMANN SURFACES AND THE HURWITZ EXISTENCE PROBLEM." Bulletin of the Australian Mathematical Society 87, no. 1 (August 2, 2012): 131–38. http://dx.doi.org/10.1017/s0004972712000469.

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AbstractThe complexity of a branched cover of a Riemann surface M to the Riemann sphere S2 is defined as its degree times the hyperbolic area of the complement of its branching set in S2. The complexity of M is defined as the infimum of the complexities of all branched covers of M to S2. We prove that if M is a connected, closed, orientable Riemann surface of genus g≥1, then its complexity equals 2π(mmin+2g−2) , where mmin is the minimum total length of a branch datum realisable by a branched cover p:M→S2.

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24

KAO, LIEN-YUNG. "Manhattan curves for hyperbolic surfaces with cusps." Ergodic Theory and Dynamical Systems 40, no. 7 (December 4, 2018): 1843–74. http://dx.doi.org/10.1017/etds.2018.124.

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In this paper, we study an interesting curve, the so-called Manhattan curve, associated with a pair of boundary-preserving Fuchsian representations of a (non-compact) surface; in particular, representations corresponding to Riemann surfaces with cusps. Using thermodynamic formalism (for countable state Markov shifts), we prove the analyticity of the Manhattan curve. Moreover, we derive several dynamical and geometric rigidity results, which generalize results of Burger [Intersection, the Manhattan curve, and Patterson–Sullivan theory in rank 2. Int. Math. Res. Not.1993(7) (1993), 217–225] and Sharp [The Manhattan curve and the correlation of length spectra on hyperbolic surfaces. Math. Z.228(4) (1998), 745–750] for convex cocompact Fuchsian representations.

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25

Mochizuki, Shinichi. "Conformal and quasiconformal categorical representation of hyperbolic Riemann surfaces." Hiroshima Mathematical Journal 36, no. 3 (November 2006): 405–41. http://dx.doi.org/10.32917/hmj/1171377082.

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26

Xie, Zheng, and Zheng Ye. "Computation of Hyperbolic Geometric Flow on Closed Riemann Surfaces." Pure and Applied Mathematics Quarterly 8, no. 4 (2012): 1075–96. http://dx.doi.org/10.4310/pamq.2012.v8.n4.a12.

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27

Ji, Lizhen. "Degeneration of pseudo-Laplace operators for hyperbolic Riemann surfaces." Proceedings of the American Mathematical Society 121, no. 1 (January 1, 1994): 283. http://dx.doi.org/10.1090/s0002-9939-1994-1184082-5.

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28

McMullen, Curtis T. "The Moduli Space of Riemann Surfaces is Kahler Hyperbolic." Annals of Mathematics 151, no. 1 (January 2000): 327. http://dx.doi.org/10.2307/121120.

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29

Dinh, Tien-Cuong, and Hao Wu. "Harmonic currents directed by foliations by Riemann surfaces." Proceedings of the American Mathematical Society 149, no. 8 (May 18, 2021): 3453–61. http://dx.doi.org/10.1090/proc/15470.

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We study local positive d d c dd^{c} -closed currents directed by a foliation by Riemann surfaces near a hyperbolic singularity which have no mass on the separatrices. A theorem of Nguyên says that the Lelong number of such a current at the singular point vanishes. We prove that this property is sharp: one cannot have any better mass estimate for this current near the singularity.

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30

HA, PHAM HOANG. "GAUSSIAN CURVATURE AND UNICITY PROBLEM OF GAUSS MAPS OF VARIOUS CLASSES OF SURFACES." Nagoya Mathematical Journal 240 (March 18, 2019): 275–97. http://dx.doi.org/10.1017/nmj.2019.5.

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In this article, we establish a new estimate for the Gaussian curvature of open Riemann surfaces in Euclidean three-space with a specified conformal metric regarding the uniqueness of the holomorphic maps of these surfaces. As its applications, we give new proofs on the unicity problems for the Gauss maps of various classes of surfaces, in particular, minimal surfaces in Euclidean three-space, constant mean curvature one surfaces in the hyperbolic three-space, maximal surfaces in the Lorentz–Minkowski three-space, improper affine spheres in the affine three-space and flat surfaces in the hyperbolic three-space.

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31

Broughton, Sean. "Simple group actions on hyperbolic Riemann surfaces of least area." Pacific Journal of Mathematics 158, no. 1 (March 1, 1993): 23–48. http://dx.doi.org/10.2140/pjm.1993.158.23.

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32

Schumacher, Georg, and Stefano Trapani. "Weil-Petersson geometry for families of hyperbolic conical Riemann surfaces." Michigan Mathematical Journal 60, no. 1 (April 2011): 3–33. http://dx.doi.org/10.1307/mmj/1301586301.

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33

Abate, Marco, and Jean-Pierre Vigu{é. "Common fixed points in hyperbolic Riemann surfaces and convex domains." Proceedings of the American Mathematical Society 112, no. 2 (February 1, 1991): 503. http://dx.doi.org/10.1090/s0002-9939-1991-1065938-8.

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34

Poletsky, Evgeny A., and Sergey E. Sharonov. "Closed Groups of Automorphisms of Products of Hyperbolic Riemann Surfaces." Journal of Geometric Analysis 28, no. 4 (December 11, 2017): 3690–707. http://dx.doi.org/10.1007/s12220-017-9972-3.

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35

Napier, T., and M. Ramachandran. "Hyperbolic Kähler manifolds and proper holomorphic mappings to Riemann surfaces." Geometric and Functional Analysis 11, no. 2 (July 2001): 382–406. http://dx.doi.org/10.1007/pl00001677.

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36

Liu, Fagui, and Yuanzhang Zhang. "Global classical solutions to hyperbolic geometric flow on Riemann surfaces." Pacific Journal of Mathematics 246, no. 2 (June 1, 2010): 333–43. http://dx.doi.org/10.2140/pjm.2010.246.333.

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37

Gradolato, Monique, and Bruno Zimmermann. "Extending finite group actions on surfaces to hyperbolic 3-manifolds." Mathematical Proceedings of the Cambridge Philosophical Society 117, no. 1 (January 1995): 137–51. http://dx.doi.org/10.1017/s0305004100072960.

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Let G be a finite group of orientation preserving isometrics of a closed orientable hyperbolic 2-manifold Fg of genus g > 1 (or equivalently, a finite group of conformal automorphisms of a closed Riemann surface). We say that the G-action on Fgbounds a hyperbolic 3-manifold M if M is a compact orientable hyperbolic 3-manifold with totally geodesic boundary Fg (as the only boundary component) such that the G-action on Fg extends to a G-action on M by isometrics. Symmetrically we will also say that the 3-manifold M bounds the given G-action. We are especially interested in Hurwitz actions, i.e. finite group actions on surfaces of maximal possible order 84(g — 1); the corresponding finite groups are called Hurwitz groups. First examples of bounding and non-bounding Hurwitz actions were given in [16].

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38

Kawakami, Yu. "Function-theoretic Properties for the Gauss Maps of Various Classes of Surfaces." Canadian Journal of Mathematics 67, no. 6 (December 1, 2015): 1411–34. http://dx.doi.org/10.4153/cjm-2015-008-5.

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AbstractWe elucidate the geometric background of function-theoretic properties for the Gauss maps of several classes of immersed surfaces in three-dimensional space forms, for example, minimal surfaces in Euclidean three-space, improper affine spheres in the affine three-space, and constant mean curvature one surfaces and flat surfaces in hyperbolic three-space. To achieve this purpose, we prove an optimal curvature bound for a specified conformal metric on an open Riemann surface and give some applications. We also provide unicity theorems for the Gauss maps of these classes of surfaces.

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39

Gušić, Dženan. "Prime Geodesic Theorems for Compact Locally Symmetric Spaces of Real Rank One." Mathematics 8, no. 10 (October 13, 2020): 1762. http://dx.doi.org/10.3390/math8101762.

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Our basic objects will be compact, even-dimensional, locally symmetric Riemannian manifolds with strictly negative sectional curvature. The goal of the present paper is to investigate the prime geodesic theorems that are associated with this class of spaces. First, following classical Randol’s appraoch in the compact Riemann surface case, we improve the error term in the corresponding result. Second, we reduce the exponent in the newly acquired remainder by using the Gallagher–Koyama techniques. In particular, we improve DeGeorge’s bound Oxη, 2ρ − ρn ≤ η < 2ρ up to Ox2ρ−ρηlogx−1, and reduce the exponent 2ρ − ρn replacing it by 2ρ − ρ4n+14n2+1 outside a set of finite logarithmic measure. As usual, n denotes the dimension of the underlying locally symmetric space, and ρ is the half-sum of the positive roots. The obtained prime geodesic theorem coincides with the best known results proved for compact Riemann surfaces, hyperbolic three-manifolds, and real hyperbolic manifolds with cusps.

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40

Verjovsky, Alberto, Shigenori Matsumoto, and Matilde Martínez. "Horocycle flows for laminations by hyperbolic Riemann surfaces and Hedlund's theorem." Journal of Modern Dynamics 10, no. 02 (May 2016): 113–34. http://dx.doi.org/10.3934/jmd.2016.10.113.

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41

Alarcón, Antonio, and José A. Gálvez. "Proper Harmonic Maps from Hyperbolic Riemann Surfaces into the Euclidean Plane." Results in Mathematics 60, no. 1-4 (August 17, 2011): 487–505. http://dx.doi.org/10.1007/s00025-011-0192-z.

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42

Bakhtin, Yuri, and Matilde Martínez. "A characterization of harmonic measures on laminations by hyperbolic Riemann surfaces." Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 44, no. 6 (December 2008): 1078–89. http://dx.doi.org/10.1214/07-aihp147.

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43

Diller, Jeffrey. "Green's functions, electric networks, and the geometry of hyperbolic Riemann surfaces." Illinois Journal of Mathematics 45, no. 2 (April 2001): 453–85. http://dx.doi.org/10.1215/ijm/1258138350.

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44

Teo, Lee-Peng. "Ruelle zeta function for cofinite hyperbolic Riemann surfaces with ramification points." Letters in Mathematical Physics 110, no. 1 (September 23, 2019): 61–82. http://dx.doi.org/10.1007/s11005-019-01222-7.

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45

Varolin, Dror. "Bergman interpolation on finite Riemann surfaces. Part II: Poincaré-Hyperbolic Case." Mathematische Annalen 366, no. 3-4 (January 19, 2016): 1137–93. http://dx.doi.org/10.1007/s00208-015-1344-3.

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46

Jorgenson, J., and R. Lundelius. "A regularized heat trace for hyperbolic Riemann surfaces of finite volume." Commentarii Mathematici Helvetici 72, no. 4 (December 1, 1997): 636–59. http://dx.doi.org/10.1007/s000140050039.

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47

Nag, Subhashis. "Self-dual connections, hyperbolic metrics and harmonic mappings on Riemann surfaces." Proceedings of the Indian Academy of Sciences - Section A 101, no. 3 (December 1991): 215–18. http://dx.doi.org/10.1007/bf02836803.

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48

JORGENSON, JAY, and LEJLA SMAJLOVIĆ. "ON THE DISTRIBUTION OF ZEROS OF THE DERIVATIVE OF SELBERG’S ZETA FUNCTION ASSOCIATED TO FINITE VOLUME RIEMANN SURFACES." Nagoya Mathematical Journal 228 (December 14, 2016): 21–71. http://dx.doi.org/10.1017/nmj.2016.52.

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We study the distribution of zeros of the derivative of the Selberg zeta function associated to a noncompact, finite volume hyperbolic Riemann surface $M$. Actually, we study the zeros of $(Z_{M}H_{M})^{\prime }$, where $Z_{M}$ is the Selberg zeta function and $H_{M}$ is the Dirichlet series component of the scattering matrix, both associated to an arbitrary finite volume hyperbolic Riemann surface $M$. Our main results address finiteness of number of zeros of $(Z_{M}H_{M})^{\prime }$ in the half-plane $\operatorname{Re}(s)<1/2$, an asymptotic count for the vertical distribution of zeros, and an asymptotic count for the horizontal distance of zeros. One realization of the spectral analysis of the Laplacian is the location of the zeros of $Z_{M}$, or, equivalently, the zeros of $Z_{M}H_{M}$. Our analysis yields an invariant $A_{M}$ which appears in the vertical and weighted vertical distribution of zeros of $(Z_{M}H_{M})^{\prime }$, and we show that $A_{M}$ has different values for surfaces associated to two topologically equivalent yet different arithmetically defined Fuchsian groups. We view this aspect of our main theorem as indicating the existence of further spectral phenomena which provides an additional refinement within the set of arithmetically defined Fuchsian groups.

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49

Chatzakos, Dimitrios, and Yiannis N. Petridis. "The hyperbolic lattice point problem in conjugacy classes." Forum Mathematicum 28, no. 5 (September 1, 2016): 981–1003. http://dx.doi.org/10.1515/forum-2015-0102.

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AbstractFor Γ a cocompact or cofinite Fuchsian group, we study the hyperbolic lattice point problem in conjugacy classes, which is a modification of the classical hyperbolic lattice point problem. We use large sieve inequalities for the Riemann surfaces ${{\Gamma\backslash{\mathbb{H}}}}$ to obtain average results for the error term, which are conjecturally optimal. We give a new proof of the error bound ${O(X^{2/3})}$, due to Good. For ${{\mathrm{SL}_{2}({\mathbb{Z}})}}$ we interpret our results in terms of indefinite quadratic forms.

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50

Yamamoto, Hiroshi. "An estimate for the hyperbolic length of closed geodesics on Riemann surfaces." Complex Variables and Elliptic Equations 64, no. 9 (December 12, 2018): 1582–607. http://dx.doi.org/10.1080/17476933.2018.1545762.

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