Decomposition of integrable holomorphic quadratic differential on Riemann surface of infinite type

2010 ◽  
Vol 53 (8) ◽  
pp. 2039-2044 ◽  
Author(s):  
Tao Cheng
Keyword(s):  
Riemann Surface ◽  
Quadratic Differential ◽  
Infinite Type ◽  
Holomorphic Quadratic Differential

PRESCRIBED HORIZONTAL AND VERTICAL TREES PROBLEM OF QUADRATIC DIFFERENTIALS

2006 ◽  
Vol 08 (03) ◽  
pp. 381-399
Author(s):  
THOMAS KWOK-KEUNG AU ◽  
TOM YAU-HENG WAN
Keyword(s):  
Riemann Surface ◽  
Quadratic Differential ◽  
Quadratic Differentials ◽  
Sufficient Condition ◽  
Holomorphic Quadratic Differential ◽  
Simply Connected ◽  
The Given

A sufficient condition for the existence of holomorphic quadratic differential on a non-compact simply-connected Riemann surface with prescribed horizontal and vertical trees is obtained. In particular, for any pair of complete ℝ-trees of finite vertices with (n + 2) infinite edges, there exists a polynomial quadratic differential on ℂ of degree n such that the associated vertical and horizontal trees are isometric to the given pair.

scholarly journals SINGULARITIES OF QUADRATIC DIFFERENTIALS AND EXTREMAL TEICHMÜLLER MAPPINGS DEFINED BY DEHN TWISTS

2009 ◽  
Vol 87 (2) ◽  
pp. 275-288 ◽  
Author(s):  
C. ZHANG
Keyword(s):  
Riemann Surface ◽  
Finite Type ◽  
Quadratic Differential ◽  
Quadratic Differentials ◽  
Closed Geodesics ◽  
Dehn Twists ◽  
Holomorphic Quadratic Differential ◽  
Simple Closed Geodesics ◽  
Disk Component ◽  
Teichmüller Mapping

AbstractLet S be a Riemann surface of finite type. Let ω be a pseudo-Anosov map of S that is obtained from Dehn twists along two families {A,B} of simple closed geodesics that fill S. Then ω can be realized as an extremal Teichmüller mapping on a surface of the same type (also denoted by S). Let ϕ be the corresponding holomorphic quadratic differential on S. We show that under certain conditions all possible nonpuncture zeros of ϕ stay away from all closures of once punctured disk components of S∖{A,B}, and the closure of each disk component of S∖{A,B} contains at most one zero of ϕ. As a consequence, we show that the number of distinct zeros and poles of ϕ is less than or equal to the number of components of S∖{A,B}.

scholarly journals The growth rate of trajectories of a quadratic differential

1990 ◽  
Vol 10 (1) ◽  
pp. 151-176 ◽  
Author(s):  
Howard Masur
Keyword(s):  
Growth Rate ◽  
Riemann Surface ◽  
Quadratic Differential ◽  
Compact Riemann Surface ◽  
Saddle Connection ◽  
Asymptotic Growth ◽  
Holomorphic Quadratic Differential ◽  
Asymptotic Growth Rate

AbstractSupposeqis a holomorphic quadratic differential on a compact Riemann surface of genusg≥ 2. Thenqdefines a metric, flat except at the zeroes. A saddle connection is a geodesic joining two zeroes with no zeroes in its interior. This paper shows the asymptotic growth rate of the number of saddles of length at mostTis at most quadratic inT. An application is given to billiards.

Measures supported on the set of uniquely ergodic directions of an arbitrary holomorphic 1-form

1999 ◽  
Vol 19 (4) ◽  
pp. 1093-1109 ◽  
Author(s):  
WILLIAM A. VEECH
Keyword(s):  
Riemann Surface ◽  
Invariant Measure ◽  
Lebesgue Measure ◽  
Quadratic Differential ◽  
Probability Measures ◽  
Locally Finite ◽  
Horocycle Flow ◽  
Closed Riemann Surface ◽  
Borel Probability ◽  
Uniquely Ergodic

We introduce a set, $Q({\bf T})$, of Borel probability measures on the circle such that each $\mu\in Q({\bf T})$ obeys the conclusion of the Kerckhoff–Masur–Smillie theorem [3]: if $q$ is a meromorphic quadratic differential with at worst simple poles on a closed Riemann surface, then for each $\mu\in Q({\bf T})$ and $\mu$-a.e. $\zeta\in{\bf T}$, $\zeta q$ has uniquely ergodic vertical foliation. As an example, the normalized Cantor–Lebesgue measure belongs to $Q({\bf T})$. The analysis also yields an analogue, for the Teichmüller horocycle flow, of a theorem of Dani: every locally finite ergodic invariant measure for the Teichmüller horocycle flow is finite.

scholarly journals On Geodesic Segments in the Infinitesimal Asymptotic Teichmüller Spaces

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Yan Wu ◽  
Yi Qi ◽  
Zunwei Fu
Keyword(s):  
Banach Space ◽  
Riemann Surface ◽  
Dual Space ◽  
Teichmüller Space ◽  
Geodesic Segment ◽  
Teichmuller Space ◽  
Quadratic Differentials ◽  
Teichmüller Spaces ◽  
Teichmuller Spaces ◽  
Infinite Type

LetAZ(R)be the infinitesimal asymptotic Teichmüller space of a Riemann surfaceRof infinite type. It is known thatAZ(R)is the quotient Banach space of the infinitesimal Teichmüller spaceZ(R), whereZ(R)is the dual space of integrable quadratic differentials. The purpose of this paper is to study the nonuniqueness of geodesic segment joining two points inAZ(R). We prove that there exist infinitely many geodesic segments between the basepoint and every nonsubstantial point in the universal infinitesimal asymptotic Teichmüller spaceAZ(D)by constructing a special degenerating sequence.

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.

Berge and Nash Equilibria in a Linear-Quadratic Differential Game

2020 ◽  
Vol 81 (11) ◽  
pp. 2108-2131
Author(s):  
V. I. Zhukovskiy ◽  
A. S. Gorbatov ◽  
K. N. Kudryavtsev
Keyword(s):  
Differential Game ◽  
Quadratic Differential ◽  
Nash Equilibria ◽  
Linear Quadratic

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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