scholarly journals Estimates of the Bergman kernel on a hyperbolic Riemann surface of finite volume-II

2020 ◽  
Vol 29 (4) ◽  
pp. 795-804
Author(s):  
Anilatmaja Aryasomayajula ◽  
Priyanka Majumder
Keyword(s):  
Riemann Surface ◽  
Finite Volume ◽  
Bergman Kernel ◽  
Hyperbolic Riemann Surface

scholarly journals ON THE DISTRIBUTION OF ZEROS OF THE DERIVATIVE OF SELBERG’S ZETA FUNCTION ASSOCIATED TO FINITE VOLUME RIEMANN SURFACES

2016 ◽  
Vol 228 ◽  
pp. 21-71 ◽  
Author(s):  
JAY JORGENSON ◽  
LEJLA SMAJLOVIĆ
Keyword(s):  
Riemann Surface ◽  
Vertical Distribution ◽  
Zeta Function ◽  
Finite Volume ◽  
Riemann Surfaces ◽  
Fuchsian Groups ◽  
Horizontal Distance ◽  
Distribution Of Zeros ◽  
Selberg Zeta Function ◽  
Hyperbolic Riemann Surface

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.

scholarly journals Examples of malformed subsets of a Riemann surface

1983 ◽  
Vol 24 (2) ◽  
pp. 101-106
Author(s):  
Moses Glasner
Keyword(s):  
Riemann Surface ◽  
Maximum Principle ◽  
Open Subset ◽  
Linear Mapping ◽  
The Maximum Principle ◽  
Hyperbolic Riemann Surface

Let R be a hyperbolic Riemann surface and W an open subset of R with ∂W piecewise analytic. Denote by the space of Dirichlet finite Tonelli functions on R and by π the harmonic projection of . Consider the relative HD–class on W, HD(W;∂W) = {u∈ │ u │ W∈HD(W) and u │ R\W = 0}. The extremization operation μis the linear mapping of HD(W;∂W) into HD(R) defined by μ. Since π preserves values of functions at the Royden harmonic boundary, the maximum principle implies that μis an order preserving injection and that Mμ is an isometry with respect to the supremum norms.

scholarly journals Analytic Functions on Some Riemann Surfaces

1963 ◽  
Vol 22 ◽  
pp. 211-217 ◽  
Author(s):  
Nobushige Toda ◽  
Kikuji Matsumoto
Keyword(s):  
Riemann Surface ◽  
Compact Subset ◽  
Analytic Functions ◽  
Riemann Surfaces ◽  
Open Riemann Surface ◽  
Interesting Theorem ◽  
Hyperbolic Riemann Surface

Some years ago, Kuramochi gave in his paper [5] a very interesting theorem, which can be stated as follows.THEOREM OF KURAMOCHI. Let R be a hyperbolic Riemann surface of the class Of OHR(OHD,resp.). Then, for any compact subset K of R such that R—K is connected, R—K as an open Riemann surface belongs to the class 0AB(OAD resp.).

scholarly journals Limits of canonical forms on towers of Riemann surfaces

2020 ◽  
Vol 2020 (764) ◽  
pp. 287-304
Author(s):  
Hyungryul Baik ◽  
Farbod Shokrieh ◽  
Chenxi Wu
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Type Theorem ◽  
Universal Cover ◽  
Canonical Forms ◽  
Classical Version ◽  
Galois Cover ◽  
Hyperbolic Form ◽  
Galois Covers ◽  
Hyperbolic Riemann Surface

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.

scholarly journals Adiabatic limit of the eta invariant over cofinite quotients of PSL(2, ℝ)

2008 ◽  
Vol 144 (6) ◽  
pp. 1593-1616 ◽  
Author(s):  
Paul Loya ◽  
Sergiu Moroianu ◽  
Jinsung Park
Keyword(s):  
Riemann Surface ◽  
Dirac Operator ◽  
Continuous Spectrum ◽  
Spin Structure ◽  
Adiabatic Limit ◽  
Regularized Trace ◽  
Standard Definition ◽  
Eta Invariant ◽  
Hyperbolic Riemann Surface

AbstractThe eta invariant of the Dirac operator over a non-compact cofinite quotient of PSL(2,ℝ) is defined through a regularized trace following Melrose. It reduces to the standard definition in terms of eigenvalues in the case of a totally non-trivial spin structure. When the S1-fibers are rescaled, the metric becomes of non-exact fibered-cusp type near the ends. We completely describe the continuous spectrum of the Dirac operator with respect to the rescaled metric and its dependence on the spin structure, and show that the adiabatic limit of the eta invariant is essentially the volume of the base hyperbolic Riemann surface with cusps, extending some of the results of Seade and Steer.

scholarly journals Banach spaces of bounded solutions of Δu = Pu (P ≥ 0) on hyperbolic riemann surfaces

1974 ◽  
Vol 53 ◽  
pp. 141-155 ◽  
Author(s):  
Mitsuru Nakai
Keyword(s):  
Banach Space ◽  
Riemann Surface ◽  
Banach Spaces ◽  
Riemann Surfaces ◽  
Space Structure ◽  
Bounded Solutions ◽  
Linear Isometry ◽  
Hölder Continuous ◽  
Hyperbolic Riemann Surfaces ◽  
Hyperbolic Riemann Surface

Consider a nonnegative Hölder continuous 2-form P(z)dxdy on a hyperbolic Riemann surface R (z = x + iy). We denote by PB(R) the Banach space of solutions of the equation Δu = Pu on R with finite supremum norms. We are interested in the question how the Banach space structure of PB(R) depends on P. Precisely we consider two such 2-forms P and Q on R and compare PB(R) and QB(R). If there exists a bijective linear isometry T of PB(R) to QB(R), then we say that PB(R) and QB(R) are isomorphic.

scholarly journals On Level Curves of Harmonic and Analytic Functions on Riemann Surfaces

1969 ◽  
Vol 34 ◽  
pp. 77-87
Author(s):  
Shinji Yamashitad
Keyword(s):  
Riemann Surface ◽  
Subharmonic Function ◽  
Analytic Functions ◽  
Harmonic Functions ◽  
Riemann Surfaces ◽  
Vector Lattice ◽  
Level Curves ◽  
Harmonic Majorant ◽  
Harmonic And Analytic Functions ◽  
Hyperbolic Riemann Surface

In this note we shall denote by R a hyperbolic Riemann surface. Let HP′(R) be the totality of harmonic functions u on R such that every subharmonic function | u | has a harmonic majorant on R. The class HP′(R) forms a vector lattice under the lattice operations:

scholarly journals Green Potential of Evans Type on Royden’s Compactification of a Riemann Surface

1964 ◽  
Vol 24 ◽  
pp. 205-239 ◽  
Author(s):  
Mitsuru Nakai
Keyword(s):  
Riemann Surface ◽  
Green Function ◽  
Green Potential ◽  
The Green Function ◽  
Hyperbolic Riemann Surface

Let R be a hyperbolic Riemann surface and gw(z) be the Green function on R with its pole w in R. We denote by (R) the totality of sequences of points in R not accumulating in R and

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