Discrete analogs of Farkas and Accola’s theorems on hyperelliptic coverings of a Riemann surface of genus 2

2014 ◽  
Vol 96 (1-2) ◽  
pp. 84-94 ◽  
Author(s):  
I. A. Mednykh
Keyword(s):  
Riemann Surface ◽  
Genus 2 ◽  
Discrete Analogs

scholarly journals RIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES

2014 ◽  
Vol 25 (01) ◽  
pp. 1450006 ◽  
Author(s):  
GAUTAM BHARALI ◽  
INDRANIL BISWAS
Keyword(s):  
Riemann Surface ◽  
Special Form ◽  
General Class ◽  
Specific Information ◽  
Complex Manifolds ◽  
Holomorphic Maps ◽  
Fiber Space ◽  
Rigidity Result ◽  
Holomorphic Map ◽  
Genus 2

In the study of holomorphic maps, the term "rigidity" refers to certain types of results that give us very specific information about a general class of holomorphic maps owing to the geometry of their domains or target spaces. Under this theme, we begin by studying when, given two compact connected complex manifolds X and Y, a degree-one holomorphic map f : Y → X is a biholomorphism. Given that the real manifolds underlying X and Y are diffeomorphic, we provide a condition under which f is a biholomorphism. Using this result, we deduce a rigidity result for holomorphic self-maps of the total space of a holomorphic fiber space. Lastly, we consider products X = X1 × X2 and Y = Y1 × Y2 of compact connected complex manifolds. When X1 is a Riemann surface of genus ≥ 2, we show that any non-constant holomorphic map F : Y → X is of a special form.

Theta constants of two kinds on a compact Riemann surface of genus 2

1970 ◽  
Vol 23 (1) ◽  
pp. 381-407 ◽  
Author(s):  
Harry E. Rauch ◽  
Hershel M. Farkas
Keyword(s):  
Riemann Surface ◽  
Compact Riemann Surface ◽  
Genus 2

A FAMILY OF SURFACES CONSTRUCTED FROM GENUS 2 CURVES

2007 ◽  
Vol 18 (05) ◽  
pp. 585-612 ◽  
Author(s):  
CHAD SCHOEN
Keyword(s):  
Riemann Surface ◽  
Two Dimensional ◽  
Hodge Conjecture ◽  
Analytic Variety ◽  
Genus 2 ◽  
Complex Analytic Variety ◽  
Projective Manifolds ◽  
Genus 2 Curves ◽  
Complex Projective

We consider the deformations of the two-dimensional complex analytic variety constructed from a genus 2 Riemann surface by attaching its self-product to its Jacobian in an elementary way. The deformations are shown to be unobstructed, the variety smooths to give complex projective manifolds whose invariants are computed and whose images under Albanese maps (re)verify an instance of the Hodge conjecture for certain abelian fourfolds.

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 On the existence of various bounded harmonic functions with given periods, II

1975 ◽  
Vol 56 ◽  
pp. 1-5
Author(s):  
Masaru Hara
Keyword(s):  
Riemann Surface ◽  
Harmonic Function ◽  
Harmonic Functions ◽  
Dirichlet Integral ◽  
Period Function ◽  
Boundedness Property ◽  
One Dimensional ◽  
Bounded Harmonic Functions

Given a harmonic function u on a Riemann surface R, we define a period functionfor every one-dimensional cycle γ of the Riemann surface R. Γx(R) denote the totality of period functions Γu such that harmonic functions u satisfy a boundedness property X. As for X, we let B stand for boundedness, and D for the finiteness of the Dirichlet integral.

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