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 TRANSVERSAL KILLING AND TWISTOR SPINORS ASSOCIATED TO THE BASIC DIRAC OPERATORS

2013 ◽  
Vol 25 (08) ◽  
pp. 1330011 ◽  
Author(s):  
ADRIAN MIHAI IONESCU ◽  
VLADIMIR SLESAR ◽  
MIHAI VISINESCU ◽  
GABRIEL EDUARD VÎLCU
Keyword(s):  
Riemannian Manifold ◽  
Dirac Operator ◽  
Mean Curvature ◽  
Spin Structure ◽  
Natural Extension ◽  
Dirac Operators ◽  
Harmonic Mean ◽  
Standard Definition ◽  
Killing Spinors ◽  
Harmonic Spinors

We study the interplay between the basic Dirac operator and the transversal Killing and twistor spinors. In order to obtain results for the general Riemannian foliations with bundle-like metric, we consider transversal Killing spinors that appear as natural extension of the harmonic spinors associated with the basic Dirac operator. In the case of foliations with basic-harmonic mean curvature, it turns out that this type of spinors coincide with the standard definition. We obtain the corresponding version of classical results on closed Riemannian manifold with spin structure, and extending some previous results.

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 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.

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