scholarly journals Canonical isomorphisms of energy finite solutions of Δu = Pu on open Riemann surfaces

1975 ◽  
Vol 56 ◽  
pp. 79-84 ◽  
Author(s):  
Mitsuru Nakai
Keyword(s):  
Riemann Surface ◽  
Continuous Function ◽  
Linear Space ◽  
Riemann Surfaces ◽  
Second Order ◽  
Local Parameter ◽  
Hölder Continuous

We call a second order differential P(z)dxdy on a Riemann surface R a density if it is not identically zero and P(z) is a nonnegative Hölder continuous function of the local parameter z = x + iy in each parametric disk. To each density P on R we associate the linear space P(R) of C2 solutions of the equation Δu(z) = P(z)u(z) invariantly defined on R. We also consider subspaces PX(R) of P(R) consisting of solutions with certain boundedness properties X.

scholarly journals An example on canonical isomorphism

1978 ◽  
Vol 70 ◽  
pp. 25-40 ◽  
Author(s):  
Mitsuru Nakai
Keyword(s):  
Riemann Surface ◽  
Second Order ◽  
Canonical Isomorphism ◽  
Hölder Continuous

A nonnegative locally Hölder continuous second order differential P = P(z)dxdy (z = x + iy) on a Riemann surface R is referred to as a density on R. A density P is said to be finite if P is integrable over R, i.e.(1) ∫ R P(z)dxdy < ∞.

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 Order Comparisons on Canonical Isomorphisms

1973 ◽  
Vol 50 ◽  
pp. 67-87 ◽  
Author(s):  
Mitsuru Nakai
Keyword(s):  
Riemann Surface ◽  
Linear Space ◽  
Hölder Continuous ◽  
Boundedness Property

Consider a nonnegative Hölder continuous 2-form P(z)dxdy (z = x + iy) on a connected Riemann surface R. We denote by P(R) the linear space of solutions u of the equation Δu = Pu on R and by PX(R) the subspace of P(R) consisting of those u with a certain boundedness property X. We also use the standard notations H(R) and HX(R) for P(R) and PX(R) with P ≡ 0.

scholarly journals Entropy for hyperbolic Riemann surface laminations I

2014 ◽  
Author(s):  
Tien-Cuong Dinh ◽  
Viet-Anh Nguyen ◽  
Nessim Sibony
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Careful Analysis ◽  
Probability Measures ◽  
Metric Entropy ◽  
Hölder Continuous ◽  
Poincaré Metric ◽  
Singular Foliations ◽  
Open Questions ◽  
Hyperbolic Riemann Surface

This chapter introduces a notion of entropy for possibly singular hyperbolic laminations by Riemann surfaces. It also studies the transverse regularity of the Poincaré metric and the finiteness of the entropy. The chapter first focuses on compact laminations, which are transversally smooth, before turning to the case of singular foliations, showing how the Poincaré metric on leaves is transversally Hölder continuous. In addition, the chapter considers the problem in the proof that the entropy is finite for singular foliations is quite delicate and requires a careful analysis of the dynamics around the singularities. Finally, the chapter discusses a notion of metric entropy for harmonic probability measures and gives some open questions.

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.

scholarly journals Padé approximants on Riemann surfaces and KP tau functions

2021 ◽  
Vol 11 (4) ◽  
Author(s):  
Marco Bertola
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Hilbert Problem ◽  
Line Bundles ◽  
Fixed Degree ◽  
Tau Function ◽  
Tau Functions ◽  
Deformation Parameters ◽  
Higher Genus ◽  
Geometric Solutions

AbstractThe paper has two relatively distinct but connected goals; the first is to define the notion of Padé approximation of Weyl–Stiltjes transforms on an arbitrary compact Riemann surface of higher genus. The data consists of a contour in the Riemann surface and a measure on it, together with the additional datum of a local coordinate near a point and a divisor of degree g. The denominators of the resulting Padé-like approximation also satisfy an orthogonality relation and are sections of appropriate line bundles. A Riemann–Hilbert problem for a square matrix of rank two is shown to characterize these orthogonal sections, in a similar fashion to the ordinary orthogonal polynomial case. The second part extends this idea to explore its connection to integrable systems. The same data can be used to define a pairing between two sequences of line bundles. The locus in the deformation space where the pairing becomes degenerate for fixed degree coincides with the zeros of a “tau” function. We show how this tau function satisfies the Kadomtsev–Petviashvili hierarchy with respect to either deformation parameters, and a certain modification of the 2-Toda hierarchy when considering the whole sequence of tau functions. We also show how this construction is related to the Krichever construction of algebro-geometric solutions.

A biregular extension theorem from a fractal surface in

2011 ◽  
Vol 18 (1) ◽  
pp. 21-29
Author(s):  
Ricardo Abreu Blaya ◽  
Juan Bory Reyes ◽  
Tania Moreno García
Keyword(s):  
Continuous Function ◽  
Bounded Domain ◽  
Extension Theorem ◽  
Fractal Surface ◽  
Fractal Boundary ◽  
Hölder Continuous

Abstract The aim of this paper is to prove the characterization on a bounded domain of with fractal boundary and a Hölder continuous function on the boundary guaranteeing the biregular extendability of the later function throughout the domain.

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