Jacobi Matrices and the Spectrum of the Neumann Operator on a Family of Riemann Surfaces

1993 ◽  
Vol 45 (4) ◽  
pp. 709-726
Author(s):  
Julian Edward
Keyword(s):  
Harmonic Function ◽  
Riemann Surfaces ◽  
Smooth Manifold ◽  
Hypergeometric Functions ◽  
Normal Derivative ◽  
Jacobi Matrices ◽  
Uniform Bounds ◽  
Neumann Operator ◽  
Direct Sum ◽  
Asymptotics Of The Eigenvalues

AbstractThe Neumann operator is an operator on the boundary of a smooth manifold which maps the boundary value of a harmonic function to its normal derivative. The spectrum of the Neumann operator is studied on the curves bounding a family of Riemann surfaces. The Neumann operator is shown to be isospectral to a direct sum of symmetric Jacobi matrices, each acting on l2(ℤ). The Jacobi matrices are shown to be isospectral to generators of bilateral, linear birth-death processes. Using the connection between Jacobi matrices and continued fractions, it is shown that the eigenvalues of the Neumann operator must solve a certain equation involving hypergeometric functions. Study of the equation yields uniform bounds on the eigenvalues and also the asymptotics of the eigenvalues as the curves degenerate into a wedge of circles.

scholarly journals Some Theorems on Open Riemann Surfaces

1951 ◽  
Vol 3 ◽  
pp. 141-145 ◽  
Author(s):  
Masatsugu Tsuji
Keyword(s):  
Riemann Surface ◽  
Harmonic Function ◽  
Green’S Function ◽  
Green's Function ◽  
Riemann Surfaces ◽  
Dirichlet Integral ◽  
Open Riemann Surface ◽  
Surface Spread

Let F be an open Riemann surface spread over the z-plane. We say that F is of positive or null boundary, according as there exists a Green’s function on F or not, Let u(z) be a harmonic function on Fand be its Dirichlet integral As R. Nevanlinna proved, if F is of null boundary, there exists no one-valued non-constant harmonic function on F5 whose Dirichlet integral is finite, This Nevanlinna’s theorem was proved very simply by Kuroda.

Riemann Surfaces, Coverings, and Hypergeometric Functions

2017 ◽  
Author(s):  
Paula Tretkoff
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Hypergeometric Functions ◽  
Euler Number ◽  
Linear Transformations ◽  
Triangle Groups ◽  
Finite Subgroups ◽  
Basic Facts ◽  
Biholomorphic Map ◽  
Simply Connected

This chapter deals with Riemann surfaces, coverings, and hypergeometric functions. It first considers the genus and Euler number of a Riemann surface before discussing Möbius transformations and notes that an automorphism of a Riemann surface is a biholomorphic map of the Riemann surface onto itself. It then describes a Riemannian metric and the Gauss-Bonnet theorem, which can be interpreted as a relation between the Gaussian curvature of a compact Riemann surface X and its Euler characteristic. It also examines the behavior of the Euler number under finite covering, along with finite subgroups of the group of fractional linear transformations PSL(2, C). Finally, it presents some basic facts about the classical Gauss hypergeometric functions of one complex variable, triangle groups acting discontinuously on one of the simply connected Riemann surfaces, and the hypergeometric monodromy group.

On an Extremal Problem Involving Harmonic Functions

1988 ◽  
Vol 31 (1) ◽  
pp. 63-69 ◽  
Author(s):  
E. J. P. Georg Schmidt
Keyword(s):  
Harmonic Function ◽  
Unique Solution ◽  
Extremal Problem ◽  
Open Subset ◽  
Harmonic Functions ◽  
Positive Measure ◽  
Normal Derivative ◽  
Regularity Conditions ◽  
Boundary Values ◽  
Almost Everywhere

AbstractGiven a domain D in R” and two specified points P0 and P1 in D we consider the problem of minimizing u(p1) over all functions harmonic in D with values between 0 and 1 normalised by the requirement u(P0) = 1/2. We show that when D is suitably regular the problem has a unique solution u* which necessarily takes on boundary values 0 or 1 almost everywhere on the boundary. In the process we prove that it is possible to separate P0 and P1by a harmonic function whose boundary value is supported in an arbitrary set of positive measure. These results depend on the fact that (under suitable regularity conditions) a harmonic function which vanishes on an open subset of the boundary has a normal derivative which is almost everywhere non-vanishing in that set.

Note on the slow motion of fluid

1936 ◽  
Vol 32 (4) ◽  
pp. 598-613 ◽  
Author(s):  
W. R. Dean
Keyword(s):  
Harmonic Function ◽  
Viscous Liquid ◽  
Stream Function ◽  
Slow Motion ◽  
Normal Derivative ◽  
Great Distance ◽  
Sharp Edge ◽  
Two Dimensional ◽  
Shearing Motion

In this paper we consider the slow two-dimensional motion of viscous liquid past a sharp edge projecting into and normal to the undisturbed direction of the stream. The liquid is supposed bounded by rigid planes represented by ABCDE in Fig. 1, and, apart from the disturbance caused by the projection, is assumed to be in uniform shearing motion. The stream function is then a bi-harmonic function that must vanish together with its normal derivative at all points of the boundary, and must be proportional to y2 at a great distance from the projection.

scholarly journals Convergence of the Weil–Petersson metric on the Teichmüller space of bordered Riemann surfaces

2016 ◽  
Vol 19 (01) ◽  
pp. 1650025 ◽  
Author(s):  
David Radnell ◽  
Eric Schippers ◽  
Wolfgang Staubach
Keyword(s):  
Riemann Surface ◽  
Tangent Space ◽  
Riemann Surfaces ◽  
Tangent Vector ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Holomorphic Curve ◽  
Hilbert Manifold ◽  
Direct Sum ◽  
Genus Formula

Consider a Riemann surface of genus [Formula: see text] bordered by [Formula: see text] curves homeomorphic to the unit circle, and assume that [Formula: see text]. For such bordered Riemann surfaces, the authors have previously defined a Teichmüller space which is a Hilbert manifold and which is holomorphically included in the standard Teichmüller space. We show that any tangent vector can be represented as the derivative of a holomorphic curve whose representative Beltrami differentials are simultaneously in [Formula: see text] and [Formula: see text], and furthermore that the space of [Formula: see text] differentials in [Formula: see text] decomposes as a direct sum of infinitesimally trivial differentials and [Formula: see text] harmonic [Formula: see text] differentials. Thus the tangent space of this Teichmüller space is given by [Formula: see text] harmonic Beltrami differentials. We conclude that this Teichmüller space has a finite Weil–Petersson Hermitian metric. Finally, we show that the aforementioned Teichmüller space is locally modeled on a space of [Formula: see text] harmonic Beltrami differentials.

Filling words in fundamental groups of Riemann surfaces

2013 ◽  
Vol 50 (1) ◽  
pp. 31-50
Author(s):  
C. Zhang
Keyword(s):  
Riemann Surface ◽  
Fundamental Group ◽  
Riemann Surfaces ◽  
Fundamental Groups ◽  
Closed Geodesics ◽  
New Words

The purpose of this article is to utilize some exiting words in the fundamental group of a Riemann surface to acquire new words that are represented by filling closed geodesics.

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