Veech surfaces and simple closed curves
نویسندگان
چکیده
We study the SL(2,R)–infimal lengths of simple closed curves on half-translation surfaces. Our main result is a characterization of Veech surfaces in terms of these lengths. We also revisit the “no small virtual triangles” theorem of Smillie and Weiss and establish the following dichotomy: the virtual triangle area spectrum of a half-translation surface either has a gap above zero or is dense in a neighborhood of zero. These results make use of the auxiliary polygon associated to a curve on a half-translation surface, as introduced by Tang and Webb.
منابع مشابه
NON-VEECH SURFACES IN Hhyp(4) ARE GENERIC
We show that every surface in the component Hhyp(4), that is the moduli space of pairs (M, ω) where M is a genus three hyperelliptic Riemann surface and ω is an Abelian differential having a single zero on M, is either a Veech surface or a generic surface, i.e. its GL(2,R)-orbit is either a closed or a dense subset ofHhyp(4). The proof develops new techniques applicable in general to the proble...
متن کاملTENSION QUARTIC TRIGONOMETRIC BÉZIER CURVES PRESERVING INTERPOLATION CURVES SHAPE
In this paper simple quartic trigonometric polynomial blending functions, with a tensionparameter, are presented. These type of functions are useful for constructing trigonometricB´ezier curves and surfaces, they can be applied to construct continuous shape preservinginterpolation spline curves with shape parameters. To better visualize objects and graphics atension parameter is included. In th...
متن کامل5 Siegel – Veech Constants in H ( 2 )
Abelian differentials on Riemann surfaces can be seen as translation surfaces, which are flat surfaces with cone-type singularities. Closed geodesics for the associated flat metrics form cylinders, whose number under a given maximal length generically has quadratic asymptotics in this length. Siegel–Veech constants are coefficients of these quadratic growth rates, and coincide for almost all su...
متن کاملPeriodic Points on Veech Surfaces and the Mordell-weil Group over a Teichmüller Curve
Periodic points are points on Veech surfaces, whose orbit under the group of affine diffeomorphisms is finite. We characterise those points as being torsion points if the Veech surfaces is suitably mapped to its Jacobian or an appropriate factor thereof. For a primitive Veech surface in genus two we show that the only periodic points are the Weierstraß points and the singularities. Our main too...
متن کاملSiegel – Veech constants in H ( 2 ) S
Abelian differentials on Riemann surfaces can be seen as translation surfaces, which are flat surfaces with cone-type singularities. Closed geodesics for the associated flat metrics form cylinders whose number under a given maximal length was proved by Eskin and Masur to generically have quadratic asymptotics in this length, with a common coefficient constant for the quadratic asymptotics calle...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2016